1. Introduction
With the continuous growth of mobile communication services and the rapid expansion of intelligent application scenarios, wireless networks are facing increasingly stringent requirements on transmission performance metrics such as coverage probability, average ergodic rate, and area spectral efficiency. Traditional single-tier cellular networks, constrained by fixed topological structures and limited spectrum resources, often suffer from insufficient coverage capability and low spectrum utilization efficiency in complex propagation environments. These limitations make it increasingly difficult for conventional architectures to satisfy the demanding quality-of-service requirements of next-generation wireless communication systems.
In this context, unmanned aerial vehicles (UAVs) have attracted considerable attention from both academia and industry. Owing to their flexible deployment, high mobility, and high probability of establishing line-of-sight (LoS) links, unmanned aerial vehicles can serve as aerial base stations, relays, or mobile edge computing nodes to enhance the coverage and capacity of terrestrial networks. The integration of unmanned aerial vehicles into heterogeneous networks (HetNets) creates a three-dimensional architecture that combines the advantages of terrestrial infrastructure and aerial nodes, thereby offering a promising solution for future ubiquitous connectivity.
Furthermore, emerging technologies such as reconfigurable intelligent surfaces (RISs), non-orthogonal multiple access (NOMA), millimeter-wave (mmWave), and terahertz (THz) communications have opened new degrees of freedom for performance enhancement. RISs can proactively manipulate the wireless propagation environment by adjusting the phase shifts of massive passive reflecting elements, thereby improving signal quality without additional spectrum consumption. NOMA enables multiple users to share the same time-frequency resources with different power levels, thus enhancing spectral efficiency and supporting massive connectivity. mmWave and THz bands provide abundant available spectrum resources, which are essential for achieving ultra-high data rates in future networks.
Motivated by these observations, this thesis focuses on modeling and analyzing the transmission performance of two types of unmanned aerial vehicles-assisted heterogeneous networks. The first scenario considers a UAV-mounted RIS to assist macro base station transmissions in a two-tier HetNet. The second scenario constructs a hybrid HetNet consisting of mmWave ground base stations employing NOMA and THz UAV base stations. By utilizing stochastic geometry theory, I derive tractable analytical expressions for key performance metrics and validate them through numerical simulations. The main contributions are summarized as follows:
First, I establish a UAV-mounted RIS-assisted heterogeneous network model where macro base stations (MBSs) and pico base stations (PBSs) are spatially distributed according to independent homogeneous Poisson point processes (HPPPs). I propose a cooperative transmission scheme between the MBS and the UAV-mounted RIS. The useful signal power of the RIS-assisted link is approximated as a Gamma distribution via the moment-matching method, and the aggregate interference power is modeled as a circularly symmetric complex Gaussian (CSCG) distribution based on the central limit theorem. I derive analytical expressions for the association probability, conditional coverage probability, overall coverage probability, and rate coverage probability.
Second, I construct a NOMA-based UAV-assisted hybrid heterogeneous network model composed of a mmWave ground base station tier and a THz UAV base station tier. Directional beamforming is employed to compensate for the severe propagation losses at high frequencies. By incorporating different path loss models and Nakagami-m fading, I derive the association probability, conditional coverage probability, average ergodic rate, and area spectral efficiency. The numerical results demonstrate that the proposed hybrid network achieves a comprehensive improvement in transmission performance.
The remainder of this thesis is organized as follows. Section 2 introduces the theoretical foundations including stochastic geometry, channel models, and key technologies. Section 3 presents the UAV-mounted RIS-assisted heterogeneous network model and its coverage performance analysis. Section 4 discusses the NOMA-based UAV-assisted hybrid heterogeneous network and its transmission performance evaluation. Finally, Section 5 concludes the thesis and outlines future research directions.
2. Theoretical Foundations and Related Technologies
This section provides the necessary theoretical background for the subsequent analysis. I first introduce the fundamentals of stochastic geometry, which serves as the primary mathematical tool for modeling the spatial randomness of network nodes. Then, I discuss the channel models adopted in this thesis, followed by an overview of key enabling technologies including directional beamforming, reconfigurable intelligent surfaces, and non-orthogonal multiple access.
2.1 Fundamentals of Stochastic Geometry
Stochastic geometry is a powerful mathematical framework for analyzing the performance of wireless networks with randomly deployed nodes. It enables the characterization of interference statistics and the derivation of network-level performance metrics in a tractable manner. Among various point process models, the Poisson point process (PPP) is the most widely used due to its analytical tractability and its ability to accurately model the spatial distribution of base stations in practical deployments.
A homogeneous Poisson point process (HPPP) of intensity $\lambda$ in the $d$-dimensional Euclidean space $\mathbb{R}^d$ satisfies the following two fundamental properties:
(1) Independence property: For any disjoint bounded sets $B_1, B_2, \ldots, B_n$, the number of points in these sets, denoted by $N(B_1), N(B_2), \ldots, N(B_n)$, are independent random variables.
(2) Poisson distribution property: The number of points in any bounded set $B$ follows a Poisson distribution with mean $\lambda |B|$, where $|B|$ denotes the Lebesgue measure of $B$:
$$P\{N(B) = k\} = e^{-\lambda |B|} \frac{(\lambda |B|)^k}{k!}, \quad k = 0, 1, 2, \ldots$$
In the context of heterogeneous cellular networks, the locations of base stations in different tiers are typically modeled as independent HPPPs. This allows the application of several important theorems, including Slivnyak’s theorem and the Campbell theorem, to simplify the analysis of network performance.
Slivnyak’s theorem states that for a Poisson point process, conditioning on a point at a specific location does not change the distribution of the remaining points. This property allows us to place a typical user at the origin without loss of generality, greatly facilitating the analysis of downlink performance.
Campbell’s theorem provides a convenient way to compute the expectation of sums over points of a PPP. For a non-negative measurable function $f: \mathbb{R}^d \rightarrow \mathbb{R}_+$, we have:
$$\mathbb{E}\left[\sum_{x \in \Phi} f(x)\right] = \lambda \int_{\mathbb{R}^d} f(x) dx$$
where $\Phi$ is a PPP of intensity $\lambda$.
Probability generating functional (PGFL) is another essential tool in stochastic geometry. For a PPP $\Phi$ with intensity $\lambda$, the PGFL is given by:
$$\mathbb{E}\left[\prod_{x \in \Phi} g(x)\right] = \exp\left(-\lambda \int_{\mathbb{R}^d} [1 – g(x)] dx\right)$$
The PGFL is particularly useful for deriving the Laplace transform of aggregate interference, which is a key step in coverage probability analysis.
2.2 Channel Models
Accurate channel modeling is crucial for evaluating the performance of wireless communication systems. In this thesis, I consider both large-scale fading and small-scale fading effects. The large-scale fading accounts for path loss and shadowing, while the small-scale fading captures the rapid fluctuations of the received signal amplitude due to multipath propagation.
Path loss model for mmWave links: The path loss between a ground base station and a user at distance $r$ can be expressed as:
$$l_m^k(r) = \begin{cases} C_m r^{-\alpha_m^L}, & \text{if } k = \text{LoS} \\ C_m r^{-\alpha_m^N}, & \text{if } k = \text{NLoS} \end{cases}$$
where $C_m = (c_v / (4\pi f_m))^2$ is the intercept factor, $f_m$ is the carrier frequency, $c_v$ is the speed of light, and $\alpha_m^L$ and $\alpha_m^N$ are the path loss exponents for LoS and NLoS links, respectively.
Path loss model for THz links: Due to the unique propagation characteristics in the THz band, the path loss includes an additional molecular absorption loss term:
$$l_t^k(r) = \begin{cases} C_t r^{-\alpha_t^L} e^{-\mu r}, & \text{if } k = \text{LoS} \\ C_t r^{-\alpha_t^N} e^{-\mu r} \beta, & \text{if } k = \text{NLoS} \end{cases}$$
where $C_t = (c_v / (4\pi f_t))^2$ is the intercept factor, $f_t$ is the THz carrier frequency, $\mu$ is the molecular absorption coefficient, and $\beta$ represents the additional scattering loss for NLoS propagation.
Small-scale fading: Throughout this thesis, I employ the Nakagami-m fading model, which provides a flexible characterization of different propagation environments. The channel power gain follows a Gamma distribution:
$$g \sim \Gamma(m, 1/m)$$
where $m$ is the Nakagami fading parameter. When $m = 1$, the Nakagami-m distribution reduces to the Rayleigh distribution, and when $m > 1$, it models LoS-dominated channels with lighter fading.
2.3 Key Technologies
Directional beamforming: To mitigate the severe path loss at mmWave and THz frequencies, directional beamforming is essential. The antenna gain of a sectored antenna model can be expressed as:
$$G(\theta) = \begin{cases} M, & \text{if } |\theta| \leq \theta_w \\ m, & \text{otherwise} \end{cases}$$
where $M$ and $m$ denote the main-lobe and side-lobe gains, respectively, and $\theta_w$ is the main-lobe beamwidth.
Reconfigurable intelligent surfaces: An RIS consists of a large number of passive reflecting elements, each capable of adjusting the phase of the incident signal. The phase shift matrix of an RIS with $N$ elements is given by:
$$\Theta = \text{diag}\left(e^{j\phi_1}, e^{j\phi_2}, \ldots, e^{j\phi_N}\right)$$
By properly optimizing the phase shifts, the reflected signals can be coherently combined at the intended receiver, thereby enhancing the desired signal power.
Non-orthogonal multiple access: NOMA is a promising multiple access technique that allows multiple users to share the same time-frequency resources by superimposing their signals in the power domain. The transmitted signal at the base station is expressed as:
$$x = \sqrt{a_f P} s_f + \sqrt{a_n P} s_n$$
where $s_f$ and $s_n$ are the signals intended for the far and near users, respectively, $a_f$ and $a_n$ are the corresponding power allocation coefficients satisfying $a_f + a_n = 1$ and $a_f > a_n$, and $P$ is the total transmit power.
2.4 Performance Metrics
Coverage probability: The coverage probability is defined as the probability that the received signal-to-interference-plus-noise ratio (SINR) exceeds a certain threshold $\tau$:
$$P_c(\tau) = \mathbb{P}(\text{SINR} > \tau)$$
Rate coverage probability: The rate coverage probability measures the probability that the achievable data rate of a user exceeds a target rate threshold:
$$P_r(w) = \mathbb{P}\left(B \log_2(1 + \text{SINR}) > w\right) = \mathbb{P}\left(\text{SINR} > 2^{w/B} – 1\right)$$
where $B$ is the system bandwidth.
Average ergodic rate: The average ergodic rate represents the long-term average of the instantaneous achievable rate over all fading states and random network topologies:
$$R = \mathbb{E}\left[\log_2(1 + \text{SINR})\right] = \frac{1}{\ln 2} \int_0^{\infty} \frac{\mathbb{P}(\text{SINR} > z)}{1 + z} dz$$
Area spectral efficiency: The area spectral efficiency quantifies the average throughput per unit area and is defined as the sum of the average ergodic rates multiplied by the base station density:
$$\text{ASE} = \sum_{i} \lambda_i R_i$$
where $\lambda_i$ represents the density of base stations in tier $i$.
3. UAV-Mounted RIS-Assisted Heterogeneous Network Modeling and Coverage Performance Analysis
3.1 Introduction
In complex propagation environments, the coverage capability of conventional heterogeneous networks is often limited by severe blockage and shadowing effects. To address this challenge, I propose a UAV-mounted RIS-assisted heterogeneous network architecture that leverages the synergy between unmanned aerial vehicles and RIS technology.
3.2 System Model
I consider a two-tier heterogeneous network where MBSs and PBSs are spatially distributed according to independent HPPPs denoted by $\Phi_m$ and $\Phi_p$, with densities $\lambda_m$ and $\lambda_p$, respectively. A UAV-mounted RIS is deployed above each MBS at a fixed altitude $H$, creating a one-to-one association between the MBS and the RIS. Therefore, the spatial distribution of UAV-mounted RISs follows the same point process as the MBSs, i.e., $\Phi_r \equiv \Phi_m$ and $\lambda_r = \lambda_m$. The transmit powers of MBSs and PBSs are denoted by $P_m$ and $P_p$, respectively.
Based on Slivnyak’s theorem, a typical user is assumed to be located at the origin. The typical user associates with the base station that provides the strongest average received power. The received power from a base station in tier $k \in \{m, p\}$ can be expressed as:
$$P_k = P_k^0 l_k(x_k)$$
where $P_k^0$ is the transmit power, $l_k(\cdot)$ denotes the large-scale path loss, and $x_k$ is the distance to the serving base station.
For the channel model, I consider both direct links and cascaded links through the UAV-mounted RIS. The signal received by the typical user from the associated MBS through the direct link and the RIS-assisted link can be jointly expressed. The key parameters used in this chapter are summarized in Table below.
| Symbol | Definition |
|---|---|
| $\Phi_m$, $\Phi_p$ | Point processes of MBSs and PBSs |
| $\Phi_r$ | Point process of UAV-mounted RISs |
| $\lambda_m$, $\lambda_p$ | Densities of MBSs and PBSs |
| $P_m$, $P_p$ | Transmit powers of MBS and PBS |
| $H$ | Altitude of the UAV-mounted RIS |
| $N$ | Number of reflecting elements on the RIS |
| $g_{m_i u}$, $g_{m_i r_n}$, $g_{r_n u}$ | Channel coefficients of the direct and cascaded links |
3.3 Signal Analysis
When the typical user is associated with an MBS, the useful signal power can be expressed as:
$$S_m = \left| \sqrt{l_{m_i u}} g_{m_i u} + \sum_{n=1}^{N} \sqrt{l_{m_i r_n}} \sqrt{l_{r_n u}} g_{m_i r_n} g_{r_n u} e^{j(\phi_{m_i r_n} + \phi_{r_n u})} \right|^2$$
By optimally adjusting the phase shifts of the RIS elements to align with the direct link, the maximum useful signal power is achieved:
$$S_m = \left| \sqrt{l_{m_i u}} g_{m_i u} + \sum_{n=1}^{N} \sqrt{l_{m_i r_n}} \sqrt{l_{r_n u}} g_{m_i r_n} g_{r_n u} \right|^2$$
Due to the complex coupling between the direct link and the cascaded link, the exact distribution of $S_m$ is difficult to obtain. To address this issue, I employ the moment-matching method to approximate $S_m$ as a Gamma random variable. Let us define the aggregated signal as:
$$S_m = \left| \sqrt{l_{m_i u}} g_{m_i u} + \sqrt{l_{m_i r}} \sqrt{l_{r u}} S_r \right|^2$$
where $S_r = \sum_{n=1}^{N} g_{m_i r_n} g_{r_n u}$ represents the sum of products of the cascaded channel gains. The first and second moments of $S_m$ can be derived as:
$$\mathbb{E}[S_m] = l_{m_i u} \mathbb{E}[|g_{m_i u}|^2] + l_{m_i r} l_{r u} \mathbb{E}[|S_r|^2]$$
$$\mathbb{E}[S_m^2] = l_{m_i u}^2 \mathbb{E}[|g_{m_i u}|^4] + 4 l_{m_i u} l_{m_i r} l_{r u} \mathbb{E}[|g_{m_i u}|^2] \mathbb{E}[|S_r|^2] + (l_{m_i r} l_{r u})^2 \mathbb{E}[|S_r|^4]$$
The first and second moments of the cascaded channel gains are given by:
$$\mathbb{E}[S_r] = N \left(\frac{\Gamma(m_h + 1/2)}{\Gamma(m_h)}\right) \left(\frac{\Gamma(m_r + 1/2)}{\Gamma(m_r)}\right)$$
$$\mathbb{E}[S_r^2] = N \left(\frac{\Gamma(m_h + 1/2)}{\Gamma(m_h)}\right)^2 \left(\frac{\Gamma(m_r + 1/2)}{\Gamma(m_r)}\right)^2 + N(N-1)$$
The shape parameter $\kappa_m$ and scale parameter $\theta_m$ of the Gamma approximation can then be obtained as:
$$\kappa_m = \frac{(\mathbb{E}[S_m])^2}{\mathbb{E}[S_m^2] – (\mathbb{E}[S_m])^2}$$
$$\theta_m = \frac{\mathbb{E}[S_m^2] – (\mathbb{E}[S_m])^2}{\mathbb{E}[S_m]}$$
For the interference analysis, the total interference at the typical user consists of two components: interference from MBSs and interference from PBSs. Based on the central limit theorem, when the number of reflecting elements $N$ is sufficiently large, the cascaded interference through RISs converges to a CSCG distribution. Consequently, the Laplace transform of the MBS interference when the typical user is associated with an MBS can be derived as:
$$\mathcal{L}_{I_m}(s) = \exp\left(-2\pi \lambda_m \int_x^{\infty} \frac{s P_m l_m(r)}{1 + s P_m l_m(r)} r dr\right)$$
where $x$ is the distance from the typical user to its serving MBS. Similarly, when the typical user is associated with a PBS, the Laplace transform of the MBS interference is:
$$\mathcal{L}_{I_m}(s) = \exp\left(-2\pi \lambda_m \int_{T_x}^{\infty} \frac{s P_m l_m(r)}{1 + s P_m l_m(r)} r dr\right)$$
where $T = (P_p/P_m)^{1/\alpha}$ accounts for the exclusion region due to the stronger received power from the serving PBS.
3.4 Association Probability and Distance Distributions
Based on the strongest average received power association strategy, the probability that the typical user associates with an MBS can be derived as:
$$A_m = \frac{\lambda_m}{\lambda_m + \lambda_p T^{2/\alpha}}$$
where $T = (P_p/P_m)^{1/\alpha}$. Similarly, the association probability with a PBS is:
$$A_p = 1 – A_m = \frac{\lambda_p T^{2/\alpha}}{\lambda_m + \lambda_p T^{2/\alpha}}$$
The PDF of the distance between the typical user and its serving MBS, conditioned on the user being associated with an MBS, is given by:
$$f_{R_m}(r) = \frac{2\pi(\lambda_m + \lambda_p T^{2/\alpha})}{A_m} r \exp\left(-\pi(\lambda_m + \lambda_p T^{2/\alpha}) r^2\right)$$
Similarly, the PDF of the distance to the serving PBS, conditioned on association with a PBS, is:
$$f_{R_p}(r) = \frac{2\pi(\lambda_p + \lambda_m T^{-2/\alpha})}{A_p} r \exp\left(-\pi(\lambda_p + \lambda_m T^{-2/\alpha}) r^2\right)$$
3.5 Coverage Performance Analysis
The overall coverage probability of the network can be expressed as the weighted sum of the conditional coverage probabilities of each tier:
$$P(\tau) = A_m P_m^C(\tau) + A_p P_p^C(\tau)$$
The conditional coverage probability of the MBS tier is derived as:
$$P_m^C(\tau) = \int_0^{\infty} \mathbb{P}\left(\frac{P_m S_m}{P_m I_m + P_p I_p} > \tau \bigg| R_m = r\right) f_{R_m}(r) dr$$
Since $S_m$ follows a Gamma distribution with shape parameter $\kappa_m$ and scale parameter $\theta_m$, the conditional coverage probability can be evaluated using the upper incomplete Gamma function:
$$P_m^C(\tau) = \int_0^{\infty} \frac{\Gamma\left(\kappa_m, \frac{\tau(\eta I_m + I_p)}{\theta_m}\right)}{\Gamma(\kappa_m)} f_{R_m}(r) dr$$
where $\eta = P_p/P_m$. By expanding the upper incomplete Gamma function and applying the derivative property of Laplace transforms, I obtain:
$$P_m^C(\tau) = \int_0^{\infty} \sum_{n=0}^{\kappa_m – 1} \frac{1}{n!} \left(-\frac{\tau}{\theta_m}\right)^n \left[\frac{d^n}{ds^n} \mathcal{L}_{I_m}\left(\frac{\tau}{\theta_m}\right) \mathcal{L}_{I_p}\left(\frac{\tau \eta}{\theta_m}\right)\right] f_{R_m}(r) dr$$
Similarly, the conditional coverage probability of the PBS tier is:
$$P_p^C(\tau) = \frac{\Gamma\left(m_p, \frac{\tau(\eta’ I_m + \theta_p I_p)}{\theta_p}\right)}{\Gamma(m_p)}$$
where $\eta’ = P_m/P_p$ and $\theta_p = l_{p_i r}/m_p$.
The rate coverage probability is defined as the probability that the instantaneous achievable rate exceeds a threshold $w$:
$$P_R(w) = A_m P_m^R(w) + A_p P_p^R(w)$$
where the conditional rate coverage probabilities are obtained by substituting $\tau = 2^{w/B} – 1$ into the coverage probability expressions.
3.6 Numerical Results and Discussion
| Parameter | Value |
|---|---|
| $\lambda_m$, $\lambda_p$ | $10^{-6}$ m$^{-2}$, $10^{-5}$ m$^{-2}$ |
| $P_m$, $P_p$ | 30 W, 1 W |
| $\alpha$ | 3 |
| $\alpha_r$ | 2 |
| $m$ | 1 |
| $N$ | 200 |
| $B$ | 100 MHz |
| $H$ | 100 m |
| $f_c$ | 28 GHz |
Figure 3.5 demonstrates the relationship between the association probability and the density ratio of PBSs to MBSs. It can be observed that as $\lambda_p/\lambda_m$ increases, the association probability of MBSs decreases, while that of PBSs increases. This trend indicates that a higher PBS density makes it more likely for users to find a nearby PBS with stronger received power.
Figures 3.6 and 3.7 illustrate the coverage probability versus the SIR threshold for different network architectures and different numbers of reflecting elements, respectively. The UAV-mounted RIS-assisted HetNet consistently outperforms both the building-mounted RIS-assisted HetNet and the conventional HetNet, especially in the medium-to-high SIR threshold region. This is because the UAV-mounted RIS can achieve higher LoS probability and better channel conditions compared to fixed building-mounted RISs. Additionally, increasing the number of reflecting elements $N$ improves the coverage probability, particularly in the high SIR regime, owing to the channel hardening effect and enhanced beamforming gain.
Figure 3.8 shows the effect of the UAV-mounted RIS altitude on the coverage probability. As the altitude $H$ increases from low to moderate values, the coverage probability improves due to the increased LoS probability. However, further increases in altitude lead to marginal gains or slight degradation, as the additional path loss from the longer propagation distance offsets the LoS benefit.
4. Transmission Performance Analysis of NOMA-Based UAV-Assisted Hybrid Heterogeneous Networks
4.1 Introduction
In this section, I investigate the transmission performance of a hybrid heterogeneous network that integrates mmWave ground base stations with NOMA and THz UAV base stations. The objective is to leverage the complementary advantages of these technologies to simultaneously enhance coverage, ergodic rate, and spectral efficiency.
4.2 System Model
I consider a two-tier hybrid heterogeneous network as illustrated in the figure below. The first tier consists of mmWave GBSs whose locations follow an HPPP $\Phi_m$ with density $\lambda_m$ and transmit power $P_m$. The second tier comprises THz ABSs (carried by unmanned aerial vehicles) at a fixed altitude $h$, whose horizontal projections follow an HPPP $\Phi_t$ with density $\lambda_t$ and transmit power $P_t$. All base stations and users are equipped with directional antenna arrays for beamforming.
In the GBS tier, NOMA is adopted to enhance spectral efficiency. I consider a two-user NOMA scenario where a fixed user is pre-associated with each GBS. The distance between the fixed user and its associated GBS is denoted by $r_f$. Based on the distance between the typical user and the serving GBS, the typical user can be classified as either a near user (if $r_m < r_f$) or a far user (if $r_m > r_f$).
The key parameters used in this chapter are summarized in the table below.
| Symbol | Definition |
|---|---|
| $\Phi_m$, $\Phi_t$ | Point processes of GBSs and ABSs |
| $\lambda_m$, $\lambda_t$ | Densities of GBSs and ABSs |
| $P_m$, $P_t$ | Transmit powers of GBS and ABS |
| $h$ | Altitude of the UAV base station |
| $a_m$, $a_n$ | Power allocation coefficients for far and near users |
| $r_f$ | Distance between the fixed user and its serving GBS |
4.3 Antenna and Channel Models
Antenna model: For both GBSs and ABSs, I employ a sectored antenna model. The antenna gain for a GBS transmitter and a user receiver can be expressed as:
$$G_m^i = \begin{cases} M_m M_u, & \text{with prob. } \frac{\theta_m}{2\pi} \frac{\theta_u}{2\pi} \\ M_m m_u, & \text{with prob. } \frac{\theta_m}{2\pi} (1 – \frac{\theta_u}{2\pi}) \\ m_m M_u, & \text{with prob. } (1 – \frac{\theta_m}{2\pi}) \frac{\theta_u}{2\pi} \\ m_m m_u, & \text{with prob. } (1 – \frac{\theta_m}{2\pi}) (1 – \frac{\theta_u}{2\pi}) \end{cases}$$
where $M_m$ and $m_m$ are the main-lobe and side-lobe gains of the GBS, $M_u$ and $m_u$ are the corresponding gains of the user, and $\theta_m$, $\theta_u$ are the beamwidths. For the serving link, the beam is assumed to be perfectly aligned, resulting in the maximum gain $G_m^0 = M_m M_u$.
Path loss model for GBS links:
$$l_m^k(r) = \begin{cases} C_m r^{-\alpha_m^L}, & \text{with prob. } P_m^L(r) \\ C_m r^{-\alpha_m^N}, & \text{with prob. } P_m^N(r) = 1 – P_m^L(r) \end{cases}$$
where the LoS probability is modeled as $P_m^L(r) = e^{-a r}$, with $a$ being a blockage-related constant.
Path loss model for ABS links:
$$l_t^k(r) = \begin{cases} C_t r^{-\alpha_t^L} e^{-\mu r}, & \text{with prob. } P_t^L(r) \\ C_t r^{-\alpha_t^N} e^{-\mu r} \beta, & \text{with prob. } P_t^N(r) \end{cases}$$
For unmanned aerial vehicles, the LoS probability also depends on the elevation angle $\theta$:
$$P_t^L(r) = \frac{1}{1 + C \exp(-b(\theta – C))}$$
where $\theta = \arctan(h/x)$ is the elevation angle, $x$ is the horizontal distance, and $b$, $C$ are environment-dependent constants. The distance between the ABS and the user relates to the horizontal distance by $r = \sqrt{x^2 + h^2}$.
Small-scale fading: I model the small-scale fading using the Nakagami-m distribution. The channel power gains are Gamma distributed:
$$g_m^k \sim \Gamma(h_m^k, 1/h_m^k), \quad g_t^k \sim \Gamma(h_t^k, 1/h_t^k)$$
where $h_m^k$ and $h_t^k$ are the Nakagami fading parameters for the GBS and ABS links, respectively.
4.4 Association Analysis
The typical user associates with the base station providing the strongest average received power. For a LoS GBS, the association probability is:
$$A_m^L = \int_0^{\infty} \bar{F}_m^N(r) \bar{F}_t^L(W_r) \bar{F}_t^N(r) f_m^L(r) dr$$
where $\bar{F}(\cdot)$ denotes the complementary cumulative distribution function (CCDF) of the nearest base station distance in a thinning PPP, and $W(\cdot)$ is the Lambert W function. Due to the complexity of the exact expressions, I apply thinning theory to decompose each tier into LoS and NLoS subsets. For instance, the CCDF of the nearest LoS GBS can be written as:
$$\bar{F}_m^L(r) = \exp\left(-2\pi \lambda_m \int_0^r P_m^L(t) t dt\right)$$
and the corresponding PDF is:
$$f_m^L(r) = 2\pi \lambda_m P_m^L(r) r \exp\left(-2\pi \lambda_m \int_0^r P_m^L(t) t dt\right)$$
Similarly, the CCDF and PDF for the nearest NLoS GBS are:
$$\bar{F}_m^N(r) = \exp\left(-2\pi \lambda_m \int_0^r P_m^N(t) t dt\right)$$
$$f_m^N(r) = 2\pi \lambda_m P_m^N(r) r \exp\left(-2\pi \lambda_m \int_0^r P_m^N(t) t dt\right)$$
For the ABS tier at altitude $h$, the CCDF of the nearest LoS ABS distance is:
$$\bar{F}_t^L(r) = \exp\left(-2\pi \lambda_t \int_0^{\sqrt{r^2 – h^2}} P_t^L(\sqrt{t^2 + h^2}) t dt\right)$$
and the PDF follows by differentiation.
4.5 Interference Analysis
Since the GBS tier and the ABS tier operate at different frequency bands, there is no inter-tier interference. Therefore, the aggregate interference at the typical user consists of two independent components.
GBS tier interference: The interference from the GBS tier when the typical user is served by a GBS can be expressed as:
$$I_m = \sum_{j \in \Phi_m \setminus \{i\}} \sum_{i=0}^{3} P_m G_m^i g_{m,j}^k l_m^k(r_j)$$
The Laplace transform of the GBS interference is given by:
$$\mathcal{L}_{I_m}(s) = \exp\left(-2\pi \lambda_m \sum_{i=0}^{3} p_m^i \int_{r_c}^{\infty} \left(1 – \frac{1}{(1 + s P_m G_m^i l_m^L(t)/h_m^L)^{h_m^L}}\right) P_m^L(t) t dt\right) \times \exp\left(-2\pi \lambda_m \sum_{i=0}^{3} p_m^i \int_{r_c}^{\infty} \left(1 – \frac{1}{(1 + s P_m G_m^i l_m^N(t)/h_m^N)^{h_m^N}}\right) P_m^N(t) t dt\right)$$
where $p_m^i$ is the probability of the $i$-th interference gain state, and $r_c$ is the exclusion radius determined by the association condition.
ABS tier interference: When the typical user is served by an ABS, the interference from other ABSs is:
$$I_t = \sum_{j \in \Phi_t \setminus \{i\}} \sum_{i=0}^{3} P_t G_t^i g_{t,j}^k l_t^k(r_j)$$
The Laplace transform takes a similar form but with the THz path loss and the ABS-specific LoS probability.
4.6 Performance Metrics
Coverage probability: The overall coverage probability of the hybrid network is:
$$P(\tau) = A_m^L P_m^L(\tau) + A_m^N P_m^N(\tau) + A_t^L P_t^L(\tau) + A_t^N P_t^N(\tau)$$
For the GBS tier, the conditional coverage probability must be analyzed separately for the near-user and far-user cases. When the typical user is a near user, both the fixed-user signal decoding and the self-signal decoding must be successful. Therefore, the conditional coverage probability is:
$$P_m^{near,k}(\tau) = \mathbb{P}\left(\gamma_{m,f \rightarrow near}^k > \tau_f, \; \gamma_{m,near}^k > \tau\right)$$
which can be evaluated using the derived Laplace transforms as:
$$P_m^{near,k}(\tau) = \int_0^{r_f} \sum_{n=0}^{h_m^k – 1} \frac{1}{n!} \left(-\frac{\tau_{near}}{\theta_m^k}\right)^n \left[\frac{d^n}{ds^n} \mathcal{L}_{I_m}\left(\frac{\tau_{near}}{\theta_m^k}\right)\right]_{s = \theta_m^k} f_{R_m^k}(r) dr$$
When the typical user is a far user, it directly decodes its own signal with the near-user signal treated as interference:
$$P_m^{far,k}(\tau) = \int_{r_f}^{\infty} \mathbb{P}\left(\gamma_{m,far}^k > \tau\right) f_{R_m^k}(r) dr$$
Average ergodic rate: The average ergodic rate of the network is expressed as:
$$R_{total} = A_m^L (R_{m,near}^L + R_{m,far}^L) + A_m^N (R_{m,near}^N + R_{m,far}^N) + A_t^L R_t^L + A_t^N R_t^N$$
where the conditional ergodic rates are obtained by integrating the corresponding coverage probabilities:
$$R_{m,near}^k = \int_0^{\infty} \frac{P_{m,near}^k(z)}{(1 + z) \ln 2} dz$$
Area spectral efficiency: The ASE of the hybrid network is:
$$\text{ASE} = \lambda_m R_m + \lambda_t R_t$$
where $R_m$ and $R_t$ are the average ergodic rates of the GBS and ABS tiers, respectively.
4.7 Numerical Results and Discussion
| Parameter | Value |
|---|---|
| $\lambda_m$, $\lambda_t$ | $5 \times 10^{-5}$ m$^{-2}$, $10 \times 10^{-5}$ m$^{-2}$ |
| $P_m$, $P_t$ | 35 W, 25 W |
| $\alpha_m^L$, $\alpha_m^N$ | 3, 4 |
| $\alpha_t^L$, $\alpha_t^N$ | 2, 4 |
| $h_m^L$, $h_m^N$ | 3, 1 |
| $h_t^L$, $h_t^N$ | 3, 1 |
| $\mu$ | 0.01 |
| $r_f$ | 40 m |
| $h$ | 40 m |
| $f_m$, $f_t$ | 28 GHz, 260 GHz |
Figure 4.5 illustrates the association probability as a function of the UAV altitude $h$. As the altitude increases, the GBS association probability first decreases and then increases, while the ABS association probability exhibits the opposite trend. This behavior is attributed to the tradeoff between the LoS probability improvement and the additional path loss as the altitude increases.
Figure 4.6 compares the GBS conditional coverage probability for OMA and NOMA with different power allocation coefficients. The OMA scheme achieves the highest coverage among all compared schemes due to the full power allocation. When the power allocation coefficient $a_m$ decreases, the GBS coverage probability decreases, especially in the high SINR region, because less power is allocated to the desired user.
Figure 4.7 compares the overall coverage probability of different network configurations. The hybrid heterogeneous network achieves significantly higher coverage probability than the single-tier GBS network. Notably, the NOMA-based hybrid network still outperforms the OMA-based single-tier network, demonstrating that the ABS-assisted deployment effectively compensates for the coverage loss introduced by NOMA.
Figure 4.8 shows the coverage probability versus the UAV altitude. The hybrid network exhibits an optimal altitude where the coverage probability is maximized, balancing the LoS probability improvement and the path loss increase. In contrast, the single-tier GBS network’s coverage remains constant regardless of altitude since it does not incorporate any aerial nodes.
Figure 4.9 presents the average ergodic rate versus the UAV altitude. The NOMA-based GBS tier consistently achieves higher ergodic rates than the OMA-based GBS tier. The ABS tier shows a peak rate at a moderate altitude, after which the rate decreases due to the increasing path loss and molecular absorption. The total network ergodic rate benefits from both the NOMA gain at the GBS tier and the ABS-assisted transmission.
Figure 4.10 demonstrates the relationship between the area spectral efficiency and the UAV altitude. The ASE decreases as the altitude increases due to the degradation of THz link quality. The NOMA curves are consistently higher than the OMA curves, confirming the spectral efficiency gain provided by NOMA. The gain initially decreases with altitude but then increases again at higher altitudes, which is attributed to the redistribution of users between the GBS and ABS tiers.
5. Conclusion and Future Work
5.1 Conclusion
In this thesis, I have systematically investigated the modeling and transmission performance analysis of two types of unmanned aerial vehicles-assisted heterogeneous networks based on stochastic geometry theory.
In the first part, I constructed a UAV-mounted RIS-assisted heterogeneous network model with the objective of improving coverage in complex propagation environments. I proposed a cooperative transmission scheme between the MBS and the UAV-mounted RIS. By employing the moment-matching method and the central limit theorem, I derived tractable approximations for the useful signal power and aggregate interference power. I then derived analytical expressions for the association probability, conditional coverage probability, overall coverage probability, and rate coverage probability. The numerical results demonstrated that the UAV-mounted RIS-assisted scheme achieves significant coverage improvements compared with conventional and building-mounted RIS-assisted HetNets, particularly under medium-to-high SIR thresholds. Furthermore, increasing the number of reflecting elements can further enhance the coverage performance.
In the second part, I established a NOMA-based UAV-assisted hybrid heterogeneous network consisting of mmWave ground base stations and THz UAV base stations. I modeled the GBS tier with NOMA and both tiers with directional beamforming. By incorporating location-dependent LoS/NLoS probability models and Nakagami-m fading, I derived analytical expressions for the association probability, conditional coverage probability, network coverage probability, average ergodic rate, and area spectral efficiency. The numerical results demonstrated that the proposed NOMA-based hybrid network achieves a comprehensive improvement in transmission performance. Specifically, NOMA enhances the average ergodic rate and area spectral efficiency by enabling spectrum sharing among multiple users, while the ABS-assisted deployment effectively compensates for the coverage loss caused by NOMA’s inter-user interference. The hybrid architecture thus achieves a synergistic improvement in both coverage and spectral efficiency.
5.2 Future Work
While this thesis provides valuable insights into the performance of unmanned aerial vehicles-assisted heterogeneous networks, several directions remain open for future investigation.
First, in this thesis, I assumed that the spatial distributions of UAVs and ground base stations are mutually independent. However, in practical deployments, UAV positions may be correlated with the coverage holes or traffic demands of ground networks. Future work could consider more realistic deployment models with spatial coupling between aerial and terrestrial nodes.
Second, I have focused primarily on coverage probability, average ergodic rate, and area spectral efficiency. Future research could extend the analysis to other important metrics such as energy efficiency, latency, reliability, and security performance to provide a more comprehensive evaluation of system performance.
Third, the limited battery capacity and endurance of unmanned aerial vehicles remain critical constraints. Future work could investigate energy-efficient deployment strategies, trajectory optimization, and wireless power transfer techniques to enhance the sustainability of UAV-assisted networks.
Fourth, I assumed ideal successive interference cancellation (SIC) in the NOMA analysis. In practical systems, SIC imperfections due to channel estimation errors and hardware limitations may significantly affect the performance. Future research could extend the analysis to incorporate non-ideal SIC and evaluate its impact on the overall network performance.
Fifth, the current analysis assumes single-antenna user terminals. Extending the model to multi-antenna users with advanced beamforming or combining techniques would be an interesting direction for future work, especially in the context of mmWave and THz communications where massive MIMO is expected to play a key role.
In summary, the research presented in this thesis provides a solid theoretical foundation for the design and analysis of unmanned aerial vehicles-assisted heterogeneous networks. It is my hope that these findings will contribute to the advancement of future wireless communication systems and inspire further research in this rapidly evolving field.