Full-Duplex Unmanned Aerial Vehicle Communications for Cellular Spectral Efficiency Enhancement Utilizing Device-to-Device Underlaying Structure

Abstract

Unmanned aerial vehicle (UAV) communications have gained recognition as a promising technology due to their unique characteristics of rapid deployment and flexible configuration. Meanwhile, device-to-device (D2D) and full-duplex (FD) technologies have emerged as promising methods for enhancing spectral efficiency and offloading traffic. One significant advantage of UAVs is their ability to partition suitable D2D pairs to increase cell capacity. In this paper, we present a novel network model in which UAVs are considered D2D pairs underlaying cellular networks, integrating FD into the communication links between UAVs to improve spectral efficiency. We then investigate a resource allocation problem for the proposed FD-UAV D2D underlaying structure model, with the objective of maximizing the system’s sum rate. Specifically, the UAVs in our model operate in full-duplex mode as D2D users (DUs), allowing the reuse of both the uplink and downlink subcarrier resources of cellular users (CUs). This optimization challenge is formulated as a mixed-integer nonlinear programming problem, known for its NP-hard and intractable nature. To address this issue, we propose a heuristic algorithm (HA) that decomposes the problem into two steps: power allocation and user pairing. The optimal power allocation is solved as a nonlinear programming problem by searching among a finite set, while the user pairing problem is addressed using the Kuhn–Munkres algorithm. The numerical results indicate that our proposed FD-MaxSumCell-HA (full-duplex UAVs maximizing the cell sum rate with a heuristic algorithm) scheme for FD-UAV D2D underlaying models outperforms HD-UAV underlaying cellular networks, with improved access rates for UAVs in FD-MaxSumCell-HA compared to HD-UAV networks.

1. Introduction

With the increase in mobile device usage and traffic, the spectrum has become increasingly limited. Given that device-to-device (D2D) communications can significantly enhance spectral efficiency by sharing spectrum resources with cellular users and effectively alleviate base station (BS) pressure through traffic offloading [1], it is considered a promising technique to address spectrum scarcity. Consequently, D2D communications in underlaying cellular networks have been widely investigated in recent years. Many important works have been focus on the resource allocation of D2D users (DUs) and cellular users (CUs) [2,3,4,5,6,7,8], which is mainly divided into three categories. The first category, like in Refs. [2,3,4], only allows DUs to reuse uplink subcarriers, which has a minimal affect on CUs. Ref. [3] considered a proportional fairness problem among users to guarantee the minimum individual user rate, and Ref. [4] aimed to improve energy efficiency while guaranteeing the required rate. The second category is downlink resource sharing for D2D [5,6]. In particular, Ref. [6] studied the balance of energy efficiency (EE) and spectral efficiency (SE) while DUs reuse the downlink subcarrier with CUs. The last category is joint uplink and downlink (JUAD) resource allocation [7,8]. Ref. [7] verified that the sum rate of JUAD is superior to that of the previous two, and Ref. [8] combined D2D communication with Non-Orthogonal Multiple Access technology to improve sum rate further. However, since D2D communication underlaying cellular networks needs to permit multiple DUs to share the same subcarrier with CUs, the mutual interference incurred by reusing the subcarrier will degrade the system capacity rather than improve it [9]. Thus, a more effective resource coordination scheme or other advanced technology needs to be developed to overcome this obstacle.

Unmanned aerial vehicle (UAV)-aided communications have been gaining more and more attention due to UAVs’ unique characteristics, such as accessing LoS connections easily and flexible deployment. The increasingly sophisticated intelligent path planning [10] and resource management technologies [11] for UAVs make their deployment in actual networks feasible. Thus, it makes sense to integrate D2D technology into UAV-aided networks. D2D is expected to play an important role by leveraging UAVs’ benefits [12,13,14,15,16], especially from the point of view of resource allocation, sum rate maximization, and coverage expansion. In Ref. [13], a UAV serves as a base station to maximize the sum rate for one device in D2D pairs, and a D2D link is used to extend coverage. Ref. [14] adds more constraints to maximize the sum rate, such as the power, altitude, location, and bandwidth of the UAV, but it only considers one D2D pair in which each device coexists in an underlaying manner. Refs. [14,15] focus on network energy harvesting aided by UAVs. Specifically, [15] considers a security system which aims to maximize secrecy energy efficiency, and ref. [14] tries to find an optimal transmit power vector which maximizes the sum rate of the system under minimum energy constraints. However, in the above maximization design, all communication links have unidirectional transmission, which may not meet the maximum capacity requirements of the 6G era of traffic explosion.

Due to the advances in self-interference (SI) cancellation techniques, full-duplex communications can be applied to cellular networks to potentially double the SE. The critical issue in full-duplex (FD) communications is their capability of canceling SI. In recent years, the SI cancellation techniques of analog, digital and antenna domains have been jointly applied to cancel SI by up to −125 dB [17,18,19], which makes FD a possible candidate 6G technology. Due to the same advantage as the two technologies above, combining D2D and FD is a effective way to further improve SE. Ref. [20] studied D2D underlaying cellular networks with FD BS to maximize the cell’s rate; however, the system capacity gains were still significantly affected by strong residual SI (RSI). Certainly, in addition to the capability for self-interference cancellation, the level of RSI was affected by the transmit power of FD devices. In particular, lower transmit power results in decreased RSI. Since device-to-device (D2D) communication involves short-distance links and typically operates at low transmit power, integrating FD technology into D2D communications is a logical choice.

In the research domain of FD-D2D underlaying cellular networks, various scenarios have been explored. References [21,22] address a basic scenario involving a single FD-D2D pair and a single CU. Notably, ref. [21] presents a closed-form approximation for the sum rate. The research expands into multi-user scenarios in [23,24,25]. In [23], both perfect and statistical Channel State Information (CSI) estimations are analyzed, leading to the development of a heuristic algorithm that maximizes the sum rate for cellular uplink sharing. This algorithm employs 2D global searching and the Kuhn–Munkres algorithm. According to the numerical results in [24], FD-D2D underlay systems achieve significantly higher capacity gains than traditional half-duplex D2D (HD-D2D) systems, provided there is sufficient SI cancellation. Furthermore, ref. [25] presents centralized and distributed power control strategies aimed at maximizing the throughput of D2D links. Additional promising methods for integrating FD-D2D include FD-D2D underlaying cellular networks with base station MIMO antennas, as discussed in [26,27]. However, previous studies have primarily focused on uplink spectrum sharing, which can result in resource wastage in extreme scenarios.

Different from previous works, leveraging the technical characteristics of UAVs, D2D, and FD, we propose the FD-MaxSumCell-HA (full-duplex UAVs maximizing the cell sum rate with a heuristic algorithm) scheme for a novel model of FD-UAV-aided networks based on D2D underlaying networks to maximize the entire cell’s sum rate, considering both uplink and downlink spectrum sharing. Specifically, the main contributions of this paper are summarized as follows:

• We address the optimization problem of maximizing the sum rate within a novel system model where UAVs, considered as D2D pairs, operate in FD mode, enabling the joint reuse of both uplink and downlink subcarrier resources of CUs. To tackle this challenge, we propose a heuristic algorithm consisting of two key steps: optimal power allocation for each potential DU-CU pair and the development of a maximum weighted matching algorithm. In the power allocation step, we simplify computational complexity through one-dimensional searching, thereby mitigating the overall complexity of the proposed scheme.
• We employ two metrics, specifically the sum rate of the cell and the access rate of D2D pairs, to evaluate the performance of the FD-MaxSumCell-HA scheme. Additionally, we introduce the FD-MaxSumCell-Rand (FD-D2D system maximizing the sum rate of the cell with random pairing) and HD-MaxSumCell-HA (HD-D2D system maximizing the sum rate of the cell with a heuristic algorithm) schemes as ideal benchmarks to evaluate the superiority of FD-MaxSumCell-HA.
• This paper examines three scenarios in a parameter study for FD-MaxSumCell-HA: In the first scenario, only uplink users are present in the cell, utilizing uplink sharing. The second scenario involves exclusively downlink users in the cell, employing downlink sharing. In the third scenario, which closely resembles real mobile network conditions, both uplink and downlink users coexist in the cell, and JUAD sharing is implemented.

The rest of the paper is organized as follows. Section 2 will introduce the system model and formulate the optimization problem for FD-MaxSumCell-HA, and the heuristic algorithm of the proposed scheme is presented in Section 3. The parameter studies and numerical results are presented in Section 4. Finally, we will present our conclusions in Section 5.

2. System Model and Formulation

2.1. System Model

Figure 1 shows a cell of UAV-aided networks based on the structure of D2D underlaying networks. The UAVs are considered DUs, and the base station (BS) is positioned at the center of the cell, whereas the DUs and CUs are distributed randomly within the cell. There are three categories of resource allocation in Figure 1. The first is that the DU operates in FD mode and reuses the subcarrier with the CU like the DU1-CU5 pair and DU2-CU3 pair, where the DU1-CU5 pair reuses the uplink resource while the DU2-CU3 pair shares downlink. The second one is a traditional scenario where, like the DU3-CU4 pair, the DU reuses the resource with the CU in half-duplex (HD) mode. The last one is an unpaired CU, which uses resource alone like CU1 and CU2. Since FD-D2D can nearly the double spectral efficiency of the DU, in this paper, we consider a system that only includes FD-UAV and the CU, and assume that the CU operates in a traditional half-duplex FDD. It is worth mentioning that the UAV in this networks model is a mooring UAV, because the better load capacity of mooring UAVs enables them to load FD communication equipment, which is not achieved by non-mooring UAVs. In addition, mooring UAVs can provide wired backhaul to the local server, which is more suitable for the capacity of FD technology.

We use DUi and CUj to represent the UAV and CU distributed in the cell, respectively, and $i\in DU=\left\{1,2,\dots ,Q\right\}$ and $j\in CU=\left\{1,2,\dots ,P\right\}$. The two different UAVs in the D2D pair are noted as $i_1$ and $i_2$, respectively. Moreover, we assume that the entire carrier resource is occupied by CUs and pre-allocated equally among them. To avoid more sever interference and a complex coordinate scheme, we only consider a “one to one” scenario, where each subcarrier can be reused by only one DU, and each DU is limited to reusing a single subcarrier.

The channels considered in this paper are those that experience path loss, slow shadowing, and fast fading. The channel gain between CUj and BS, denoted as $g_{j,B}$, for instance, is modeled as

$$g_{j,B}=G{\beta }_{j,B}{\mathrm{\Gamma }}_{j,B}{l}_{j,B}^{-\alpha }$$

(1)

where ${\beta }_{j,B}$ represents the gain from fast fading, which follows an exponential distribution; $\mathrm{\Gamma }j,B$ denotes the gain from slow fading, characterized by a log-normal distribution; G is the path loss constant; $\alpha$ is the path loss exponent; and $lj,B$ is the distance between CUj and the BS. Similarly, the gains of the other channels shown in Figure 1 are denoted as $g_{B,j}$, $g_{i_1,B}$, $g_{i_2,B}$, $g_{B,i_1}$, $g_{B,i_2}$, $g_{j,i_1}$, $g_{j,i_2}$, $g_{i_1,j}$, $g_{i_2,j}$, and $g_{i,i}$. In particular, the reversible links between two devices in a D2D pair are transmitted at the same frequency and same time; thus, the gains of the two directions are all denoted as $g_{i,i}$. It is common knowledge that imperfect Channel State Information (CSI) can degrade system performance. However, as the imperfect CSI did not alter the performance order in the comparison of the proposed scheme, we assume, for convenience, that the BS possesses perfect CSI for all the involved links. The definitions of the channel-related parameters are summarized in

Table 1. Definitions of channel-related parameters.

Notation	Definition
$g_{B,j}$, $g_{i_1,B}$, $g_{i_2,B}$, $g_{B,i_1}$, $g_{B,i_2}$, $g_{j,i_1}$, $g_{j,i_2}$, $g_{i_1,j}$, $g_{i_2,j}$	The channel gain between the base station, CU, and DU. The subscript B represents the base station, j represents the CU, and $i_1$ and $i_2$ represent the DU.
$\beta$	The exponential distribution coefficient of fast fading.
$\mathrm{\Gamma }$	The log-normal distribution coefficient of slow fading.
G	The constant coefficient of path loss.
$\alpha$	The exponent coefficient of path loss.
l	The distance between the CU, DU, and BS.

. In practice, typical G2A channel models, including probabilistic LoS/NLoS models and 3GPP-suggested specifications, usually consider blockage distribution to determine LoS or NLoS conditions [10]. However, to quickly validate the network structure proposed in this paper, we made a trade-off between mathematical complexity and accuracy, so our adopted path loss model does not consider this aspect.

To pair DUi and CUj, the binary variable ${\rho }_{i,j}$ is defined as a paired factor. If DUi reuses the same subcarrier (whether using the uplink or downlink) with CUj, then ${\rho }_{i,j}=1$; otherwise, ${\rho }_{i,j}=0$, ${\rho }_{i,j}^u$ is for uplink, while ${\rho }_{i,j}^d$ is for downlink. As shown in Figure 1, the interference scenario in FD-D2D systems is more complex than in traditional systems because of the residual SI of FD. We categorize the problem into two distinct cases: reusing the uplink and downlink of the CU. In the case of reusing the uplink, the SINR of CUj and the SINR of DUi can be expressed as

$$\begin{array}{cc}\hfill & {\gamma }_j^u=\frac{p_j{g}_{j,B}}{{\displaystyle \sum _{i=1}^Q}{\rho }_{i,j}^u{p}_i(g_{i_1,B}+g_{i_2,B})+N_0}\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill & {\gamma }_{i_1}^u=\frac{p_i{g}_{i,i}}{{\displaystyle \sum _{j=1}^P}{\rho }_{i,j}^u(p_j{g}_{j,i_1}+p_i·\eta )+N_0}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & {\gamma }_{i_2}^u=\frac{p_i{g}_{i,i}}{{\displaystyle \sum _{j=1}^P}{\rho }_{i,j}^u(p_j{g}_{j,i_2}+p_i·\eta )+N_0}\hfill \end{array}$$

(4)

where $p_j$ and $p_i$ denote the transmit power of CUj and DUi, respectively; $\eta$ denotes the capability of self-interference suppression (SIS); and $N_0$ is the variance of zero mean Additive White Gaussian Noise. As for the case of reusing the downlink, the SINR of CUj and DUi can be, respectively, given by

$$\begin{array}{cc}\hfill & {\gamma }_j^d=\frac{p_{B,j}{g}_{B,j}}{{\displaystyle \sum _{i=1}^Q}{\rho }_{i,j}^d{p}_i(g_{i_1,j}+g_{i_2,j})+N_0}\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {\gamma }_{i_1}^d=\frac{p_i{g}_{i,i}}{{\displaystyle \sum _{j=1}^P}{\rho }_{i,j}^d(p_{B,j}{g}_{B,i_1}+p_i·\eta )+N_0}\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & {\gamma }_{i_2}^d=\frac{p_i{g}_{i,i}}{{\displaystyle \sum _{j=1}^P}{\rho }_{i,j}^d(p_{B,j}{g}_{B,i_2}+p_i·\eta )+N_0}\hfill \end{array}$$

(7)

where $P_{B,j}$ stands for the power transmitted from the base station to CUj.

Hence, we can express the achievable rates for the uplink and downlink of CUj and its corresponding DUi as follows:

$$\begin{array}{cc}\hfill & R_j^u={log}_2(1+{\gamma }_j^u)\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & R_j^d={log}_2(1+{\gamma }_j^d)\hfill \end{array}$$

(9)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{R}_{i_1}^u={log}_2(1+{\gamma }_{i_1}^u),R_{i_2}^u={log}_2(1+{\gamma }_{i_2}^u)$$

(10)

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{R}_{i_1}^d={log}_2(1+{\gamma }_{i_1}^d),R_{i_2}^d={log}_2(1+{\gamma }_{i_2}^d)$$

(11)

And the sum rate of the overall cell is

$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{R}_{\mathrm{sum}}=\sum _{j=1}^P{R}_j^u+\sum _{j=1}^P{R}_j^d+\sum _{i=1}^Q{R}_i^u+\sum _{i=1}^Q{R}_i^d$$

(12)

where $R_i^u$ is the sum of $R_{i_1}^u$ and $R_{i_2}^u$, and $R_i^d$ is the sum of $R_{i_1}^d$ and $R_{i_2}^d$.

2.2. Problem Formulation

We investigate a resource allocation problem to maximize the sum rate of the overall system includes FD-D2D only while guaranteeing the quality of service (QoS) of both CUs and DUs. Thus, the optimization problem is presented as follows:

$$\begin{array}{cc}\hfill \mathcal{P}1:& \underset{{\rho }_{i,j},p}{\max }{R}_{\mathrm{sum}}\hfill \end{array}$$

(13)

$$\begin{array}{}\mathrm{(13a)}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.& {\gamma }_j^u\ge {\gamma }_j^{u,\mathrm{req}},{\gamma }_j^d\ge {\gamma }_j^{d,\mathrm{req}},\forall j\in \mathcal{C}\hfill \mathrm{(13b)}& & {\gamma }_{i_1}^u\ge {\gamma }_i^{\mathrm{req}},{\gamma }_{i_2}^u\ge {\gamma }_i^{\mathrm{req}}\forall i\in \mathcal{D}\hfill \mathrm{(13c)}& & {\gamma }_{i_1}^d\ge {\gamma }_i^{\mathrm{req}},{\gamma }_{i_2}^d\ge {\gamma }_i^{\mathrm{req}}\forall i\in \mathcal{D}\hfill \mathrm{(13d)}& & 0\le p_i\le p_i^{\max },\forall i\in \mathcal{D}\hfill \mathrm{(13e)}& & 0\le p_j\le p_j^{\max },\forall j\in \mathcal{C}\hfill \mathrm{(13f)}& & 0\le p_{B,j}\le p_{B,j}^{\max },\forall j\in \mathcal{C}\hfill \mathrm{(13g)}& & {\displaystyle \sum _{i=1}^Q}{\rho }_{i,j}^d+{\rho }_{i,j}^u\le 1,\forall j\in \mathcal{C}\hfill \mathrm{(13h)}& & {\displaystyle \sum _{j=1}^P}{\rho }_{i,j}^d+{\rho }_{i,j}^u\le 1,\forall i\in \mathcal{D}\hfill \mathrm{(13i)}& & {\rho }_{i,j}^d,{\rho }_{i,j}^u\in \{0,1\},\forall i\in \mathcal{D},\forall j\in \mathcal{C}\hfill \end{array}$$

where $p$ is the transmit power set including $p_i$, $p_j$, and $p_{B,j}$. In $\mathcal{P}1$, constraints (13a–c) guarantee that the data rate of CUs and DUs is above the requirements, which satisfies the QoS. ${\gamma }_j^{u,\mathrm{req}}$, ${\gamma }_j^{d,\mathrm{req}}$, and ${\gamma }_i^{\mathrm{req}}$ denote the minimum SINR requirement of the uplink and downlink for CUs and DUs, respectively. (13d–f) are the power constraints, where $p_i^{\max }$, $p_j^{\max }$, and $p_{B,j}^{\max }$ are the maximum transmit power of DUi, CUj, and BS, respectively. The “one to one” reusing scenario is ensured by (13g,h), of which (13g) ensures that each subcarrier of CUj can be reused by only one DU, and Equation (13h) guarantees that any DUj can reuse at most one subcarrier of CUs.

As the network structure we proposed is a distributed system, it is assumed that the base station knows all the channel information for the calculations. The base station solves the optimization problem we modeled by using the algorithm we designed to perform power control and resource allocation for all DUs and CUs.

In practice, in actual network deployment, besides the data channels described in Figure 1, there are also control channels. DUs and CUs periodically upload CSI to the central base station via these control channels. Additionally, CUs can also transmit CSI to the base station through the data channel while performing the uplink service. It is worth mentioning that the denser the control channel’s period, the more accurate the CSI the base station possesses. However, this also increases the system overhead and signaling interference. Conversely, the sparser the control channel’s period, the lower the system overhead and signaling interference, but the CSI might not be updated promptly. If the CSI reporting period is too long, it can lead to inaccurate calculations by the base station, as the channel gain may have undergone random changes. Fortunately, due to the short-range communication characteristic of D2D, the CSI between two DUs in D2D pairs changes relatively slowly, allowing the control channel to be set with a larger period.

3. The Proposed Heuristic Algorithm

The problem $\mathcal{P}1$ is an MINLP, which is NP-hard and mathematically intractable. Therefore, we proposed a heuristic algorithm to decompose the $\mathcal{P}1$ into two subproblems to make the MINLP tractable, i.e., the power allocation and user pairing. First, the optimal power solution for each DUi matching each CUi is given by formulating the power allocation problem as nonlinear programming and searching for the optimal solution among a finite set. If the power solution can not only make the rate of DUi and CUj satisfy the QoS, but also improve the sum rate of the DUi-CUj pair compared with CUj, the reusing pair, DUi-CUj, will be regarded as a candidate option for the user pairing subproblem. Otherwise, it will be removed from the feasible option list. Then, we need to chose the most appropriate DU-CU pairs among the feasible candidates through maximum weight bipartite matching so that the sum rate of overall system can be maximized.

3.1. Power Allocation

To search for the optimal transmit power solution of each DU-CU pair, we simplify the problem $\mathcal{P}1$ to formulate an optimized problem $\mathcal{P}2$ which considers only one DU and one CU. The optimal objective is to maximize the rate of one DU-CU pair. For instance, when DUi reuses the uplink subcarrier of CUj,  $\mathcal{P}2$ is given as

$$\begin{array}{cc}\hfill \mathcal{P}2:& \underset{p_i,p_j}{\max }{R}_{i,j}^u\hfill \end{array}$$

(14)

$$\begin{array}{}\mathrm{(14a)}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}s.t.& {\gamma }_j^u\ge {\gamma }_j^{u,\mathrm{req}}\hfill \mathrm{(14b)}& & {\gamma }_{i_1}^u\ge {\gamma }_i^{\mathrm{req}},{\gamma }_{i_2}^u\ge {\gamma }_i^{\mathrm{req}}\hfill \mathrm{(14c)}& & 0\le p_i\le p_i^{\max }\hfill \mathrm{(14d)}& & 0\le p_j\le p_j^{\max }\hfill \end{array}$$

where $R_{i,j}^u=R_j^u+R_{i_1}^u+R_{i_2}^u$, which indicates the sum rate of the pair DUi-CUj. It is evident that $\mathcal{P}2$ is a nonlinear programming problem that can be solved using geometric programming techniques. Since D2D is a type of short-range communication, to reduce the computational complexity, we set $g_{j,i_1}=g_{j,i_2}=g_{j,i}$, where $g_{j,i}$ is defined as the channel gain from CUj to the middle of two devices in D2D. Thus, we can obtain ${\gamma }_{i_1}^u={\gamma }_{i_2}^u={\gamma }_i^u$ using (2)–(4), and $R_{i_1}^u=R_{i_2}^u=R_i^u$, where ${\gamma }_i^u$ is regarded as the SINR of D2D when reusing the uplink subcarrier of CU. As can be seen in Figure 2, $l_1$ is ${\gamma }_j^u={\gamma }_j^{u,\mathrm{req}}$, $l_2$ is ${\gamma }_i^u={\gamma }_i^{\mathrm{req}}$, $l_3$ is $p_j=p_j^{\max }$, and $l_4$ is $p_i=p_i^{\max }$, the region $\mathcal{R}$ delineates the feasible power allocation space for CUj and DUi.

When searching for the optimal power solution ($p_i$, $p_j$), we introduce the following lemmas.

Theorem 1.

In the optimal power solution, at least one component must be at its maximum value. Specifically, the optimal solution ($p_i^{op}$, $p_j^{op}$) will have either $p_i^{op}=p_i^{\max }$ or $p_j^{op}=p_j^{\max }$.

Proof. 

Lemma 1 is proven by contradiction. $\mathcal{R}$ is a closed set like in Figure 2b–d or an empty set as in Figure 2a according to constraints in (14). For nonempty $\mathcal{R}$, the optimal power solution ($p_i^{op}$, $p_j^{op}$) obviously falls in $\mathcal{R}$, and it is assumed that $p_i^{op}$ and $p_j^{op}$ are below the maximum value. Then, if we substitute ($\alpha p_i^{op}$, $\alpha p_j^{op}$) for ($p_i^{op}$, $p_j^{op}$) in the objective function of $\mathcal{P}2$, in which $\forall \alpha >1$, $\alpha \in R^{+}$, we can obtain

$$\begin{array}{cc}\hfill R_{i,j}^u(\alpha p_i^{op},\alpha p_j^{op})& =R_j^u(\alpha p_i^{op},\alpha p_j^{op})+2·R_i^u(\alpha p_i^{op},\alpha p_j^{op})\hfill \\ \hfill & =lo{g}_2[(1+\frac{p_j^{op}{g}_{j,B}}{p_i^{op}(g_{i_1,B}+g_{i_2,B})+(N_0/\alpha )})\times \hfill \\ \hfill & {(1+\frac{p_i^{op}{g}_{i,i}}{p_j^{op}{g}_{j,i}+p_i^{op}·\eta +(N_0/\alpha )})}^2]\hfill \\ \hfill & >R_{i,j}^u(p_i^{op},p_j^{op}).\hfill \end{array}$$

(15)

Using (15), we obtain $R_{i,j}^u(\alpha p_i^{op}$, $\alpha p_j^{*}op)>R_{i,j}^u(p_i^{op},p_j^{op})$, while $\alpha >1$. This obviously contradicts the assumption that ($p_i^{op}$, $p_j^{op}$) is the best possible solution. Thus, at least one component of the optimal solution $(p_i^{op}$, $p_j^{op})$ has to reach the maximum value $p_i^{\max }$ or $p_j^{\max }$.   □

Lemma 1 illustrates that the optimal solution lies at the boundaries of the feasible region. As Figure 2 shows, there are four possible scenarios for the feasible region $\mathcal{R}$, which depend on different maximum transmit power levels, channel gains, and SINR requirements [2]. The most favorable solution exists at the line $\overline{Z_1{Z}_2}$, $\overline{Z_2{Z}_3}$, $\overline{Z_3{Z}_4}$ or the line $\overline{Z_1{Z}_5}$ in Figure 2. To further find the collection of potential optimal power solutions, we introduce Lemma 2 as follows.

Theorem 2.

If the feasible region $\mathcal{R}$ is limited, the most favorable solution $(p_i^{op},p_j^{op})$ can only exist at the corners of $\mathcal{R}$.

Proof. 

Let $\partial \mathcal{R}$ denote the boundary of $\mathcal{R}$. The region $\mathcal{R}$ is enclosed by four lines, which are $l_1$, $l_2$, $l_3$, and $l_4$. According to the conclusion of Lemma 1, we need to search for extreme points of objective function on $\partial \mathcal{R}$. Lemma 2 is demonstrated for the following cases:

(1)

If the geometric programming situation is as in Figure 2c, $(p_i^{op},p_j^{op})$∈$\overline{Z_3{Z}_4}$. Since $R_{i,j}^u$ is a convex function [28], we have $\frac{{\partial }^2{R}_{i,j}^u}{\partial {p_i}^2}\ge 0$; thus, the optimal solution can only exist at points $Z_3$ and $Z_4$.

(2)

If the geometric programming situation is as in Figure 2d, $(p_i^{op},p_j^{op})$∈$\overline{Z_1{Z}_5}$. Since $R_{i,j}^u$ is a convex function, we have $\frac{{\partial }^2{R}_{i,j}^u}{\partial {p_j}^2}\ge 0$; thus, the optimal solution can only exist at points $Z_1$ and $Z_5$.

(3)

If the geometric programming situation is as in Figure 2b, ($p_i$, $p_j$) ∈$\overline{Z_1{Z}_2}$ and $\overline{Z_2{Z}_3}$. Similar to (1) and (2), the optimal solution can only exist at points $Z_1$, $Z_2$, and $Z_3$.

Therefore, we conclude that the optimal solution $(p_i^{op},p_j^{op})$ can only exist at the vertices of region $\mathcal{R}$.    □

Based on the above lemmas, the possible objective points for the optimal power solution are indicated in Figure 2, which are $Z_1$ to $Z_5$. The coordinates of $Z_0$ and the slope of $l_1$ and $l_2$ determine whether there are solutions or not, which is illustrated in ref. [2]. We notate points $Z_0(p_j^{Z_0},p_i^{Z_0})$, $Z_1(p_j^{Z_1},p_i^{\max })$, $Z_2(p_j^{\max },p_i^{\max })$, $Z_3(p_j^{\max },p_i^{Z_3})$, $Z_4(p_j^{\max },p_i^{Z_4})$, and $Z_5(p_j^{Z_5},p_i^{\max })$. Since $Z_1$ to $Z_5$ are at the intersection of lines $l_1$, $l_2$, $l_3$, and $l_4$, we can obtain the values of $p_j^{Z_0}$, $p_i^{Z_0}$, $p_j^{Z_1}$, $p_i^{Z_3}$, $p_i^{Z_4}$, and $p_j^{Z_5}$ as follows:

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} P_j^{Z_0}=& \frac{{\gamma }_j^{u,\mathrm{req}}{N}_0(g_{i,i}-{\gamma }_i^{\mathrm{req}}\eta )+{\gamma }_j^{u,\mathrm{req}}{\gamma }_i^{\mathrm{req}}{N}_0(g_{i_1,B}+g_{i_2,B})}{g_{j,B}(g_{i,i}-{\gamma }_i^{\mathrm{req}}\eta )-{\gamma }_j^{u,\mathrm{req}}{\gamma }_i^{\mathrm{req}}{g}_{i,i}(g_{i_1,B}+g_{i_2,B})}\hfill \end{array}$$

(16)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} P_i^{Z_0}=& \frac{{\gamma }_j^{u,\mathrm{req}}{N}_0(g_{i,i}-{\gamma }_i^{\mathrm{req}}\eta )+{\gamma }_j^{u,\mathrm{req}}{\gamma }_i^{\mathrm{req}}{N}_0(g_{i_1,B}+g_{i_2,B})}{g_{j,B}(g_{i,i}-{\gamma }_i^{\mathrm{req}}\eta )-{\gamma }_j^{u,\mathrm{req}}{\gamma }_i^{\mathrm{req}}{g}_{i,i}(g_{i_1,B}+g_{i_2,B})}\hfill \\ \hfill & \times \frac{{\gamma }_i^{\mathrm{req}}{g}_{i,i}}{g_{i,i}-{\gamma }_i^{\mathrm{req}}\eta }+\frac{{\gamma }_i{N}_0}{g_{i,i}-{\gamma }_i^{\mathrm{req}}\eta }\hfill \end{array}\end{array}$$

(17)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{P}_j^{Z_1}=& \frac{{\gamma }_j^{u,\mathrm{req}}[P_i^{\max }(g_{i_1,B}+g_{i_2,B})+N_0]}{g_{j,B}}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill P_i^{Z_3}=& \frac{P_j^{\max }{g}_{j,B}-{\gamma }_j^{u,\mathrm{req}}{N}_0}{g_{i,i}-{\gamma }_i^{\mathrm{req}}\eta }\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill P_i^{Z_4}=& \frac{P_j^{\max }{g}_{j,B}-{\gamma }_j^{u,\mathrm{req}}{N}_0}{{\gamma }_j^{u,\mathrm{req}}{g}_{i,B}}\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill \hspace{1em}\hspace{1em}\hspace{1em}{P}_j^{Z_5}=& \frac{P_i^{\max }(g_{i,i}-{\gamma }_i^{\mathrm{req}}\eta )-{\gamma }_i^{\mathrm{req}}{N}_0}{{\gamma }_j^{u,\mathrm{req}}{g}_{i,i}}\hfill \end{array}$$

(21)

Based on the above, we obtain a finite set $\{Z_1,Z_2,Z_3,Z_4,Z_5\}$, which contains the optimal solution, so that it can be searched and compared for all elements to obtain the maximum $R_{i,j}^u$. Thus, the power allocation in the DUi-CUj pair for reusing the uplink subcarrier is solved. Similarly, for the downlink, the power allocation for the maximum $R_{i,j}^d$ can be solved by the same algorithm.

3.2. User Pairing

We proposed the most favorable power allocation algorithm for each DU-CU pair and obtained the maximal rate $R_{i,j}^u$. However, not every CU has a shared DU. For each unpaired CUj (uplink, for instance), the maximum achieved rate is

$$R_j^{u,\max }={log}_2(1+\frac{p_j^{\max }{g}_{j,B}}{N_0})$$

(22)

When an unpaired CUj shares its uplink subcarrier with DUi, the sum rate will vary. To express the rate variety, we define the cell’s capacity gain for uplink as

$$\Delta R_{i,j}^u=R_{i,j}^u-R_j^{u,\max }$$

(23)

Similarly, the cell’s capacity gain for the downlink can be defined as $\Delta R_{i,j}^d=R_{i,j}^d-R_j^{d,\max }$. Obviously, the optimal user pairing problem becomes a bipartite matching problem for reaching the maximum weight. $\mathcal{P}$3 can be formulated as

$$\begin{array}{cc}\hfill \mathcal{P}3:& \underset{{\rho }_{i,j}}{\max }\sum _{j=1}^P\sum _{i=1}^Q({\rho }_{i,j}^u\Delta R_{i,j}^u+{\rho }_{i,j}^d\Delta R_{i,j}^d)\hfill \end{array}$$

(24)

$$\begin{array}{}\mathrm{(24a)}& \hfill \hspace{1em}\hspace{1em}s.t.& {\displaystyle \sum _{j=1}^P}{\rho }_{i,j}^d+{\rho }_{i,j}^u\le 1,\forall i\in \mathcal{D}\hfill \mathrm{(24b)}& & {\displaystyle \sum _{i=1}^Q}{\rho }_{i,j}^d+{\rho }_{i,j}^u\le 1,\forall j\in \mathcal{C}\hfill \mathrm{(24c)}& & {\rho }_{i,j}^u,{\rho }_{i,j}^d\in \{0,1\},\forall i\in \mathcal{D},\forall j\in \mathcal{C}\hfill \end{array}$$

To solve $\mathcal{P}3$ through bipartite graph matching, we establish two sets of vertices; one is the set of DUs, andthe other is the set of CUs with subcarriers including the uplink and downlink. And then, we compute the weight of the edge between the two vertices with $\Delta R_{i,j}^u$ or $\Delta R_{i,j}^d$, which depends on the transmission direction of the CU. This problem is solved by Kuhn–Munkres algorithm. The specific algorithm flow for user pairing is detailed in Algorithm 1.

Algorithm 1 The optimal user pairing algorithm of HA

1: Initialize the cell’s sum rate variation matrix ${\{\Delta R_{i,j}\}}_{Q\times P}$, and the pairing indicator matrix ${\left\{{\rho }_{i,j}\right\}}_{Q\times P}$.   2: for $j=1:Q$ do   3:    for $i=1:P$ do   4:      Determine the optimal power solution $(p_i^{op},p_j^{op})$ for the single pair CUj-DUi by applying the power control algorithm described in Section 3.1.   5:      Substitute $(p_i^{op},p_j^{op})$ in Equations (2)–(10), (22) and (23) to obtain $\Delta R_{i,j}$, which includes both uplink and downlink rates.   6:      Set ${\rho }_{i,j}=1$.   7:      if $\Delta R_{i,j}<0$ then   8:         Under these conditions, we assume that the pairing attempt between CUj and DUi fails, FD-D2D access to the cell is prohibited, and CUj maintains its original connection, that is, set               $\Delta R_{i,j}=0$               $R_{i,j}^C=R_j^{unp}$               $R_{i,j}^D=0$               ${\rho }_{i,j}=0$   9:      end if 10:    end for 11: end for 12: Use the Kuhn–Munkres algorithm for maximum weight to determine the most favorable pattern ${\left\{{\rho }_{i,j}\right\}}_{Q\times P}$ of ${\{\Delta R_{i,j}\}}_{Q\times P}$. 13: Return the optimal user pairing pattern ${\left\{{\rho }_{i,j}\right\}}_{Q\times P}$ and sum of the corresponding selected elements in $\Delta R_{i,j}$.

The computational complexity of our approach is polynomial and depends on the number of vertices and edges. Specifically, the most favorable power solution for a single CU-DU pair is searched in a limited set through one-dimensional searching, in which the complexity is $\mathcal{O}\left(1\right)$. This leads to a total complexity of $\mathcal{O}\left(PQ\right)$ for the power control algorithm applied to all CU-DU pairs. Additionally, since our assumption is that the quantity of CUs is greater than or equal to the quantity of DUs, i.e., $P\ge Q$, the Kuhn–Munkres algorithm for resource allocation addresses user pairing in the complexity of $\mathcal{O}\left(P^3\right)$. Thus, the total complexity of MaxCU-OPOP is $\mathcal{O}(PQ+P^3)$, which is a significant reduction compared to the complexity recorded in refs. [23,24,25].

4. Numerical Result

In this section, the numerical result is presented to verify the proposed FD-MaxSumCell-HA scheme. We consider a circular cell with the BS located in the center, where the FD-DUs and CUs are distributed randomly. The FD-MaxSumCell-HA scheme is implemented using Monte Carlo methods over 10,000 times to smooth the randomness in the simulation. The relevant parameters in our simulation are shown in

Table 2. The parameter values used in the simulation.

Parameter	Value
Number of CUs (P)	20
Number of DUs (Q)	0 to 20
Cell radius	500 m
Users distribution	Uniform
Fast fading	Mean = 1
Slow fading	Standard deviation = 8 dB
Noise spectral density ($N_0$)	−174 dBm/Hz
$P_j^{\max }$, $P_i^{\max }$	24 dBm
$P_{BS}^{\max }$	46 dBm
Exponent coefficients of path loss ($\alpha$)	3
Constant coefficients of path loss (G)	${10}^{-2}$
D2D distance (d)	10 m
UAV hover height	80 m
Bandwidth	10 MHz
Self-interference suppression	110 dB
Number of subcarriers (L)	20
${\gamma }_j^{d,\mathrm{req}}$,${\gamma }_j^{u,\mathrm{req}}$,${\gamma }_i^{\mathrm{req}}$	10 dB

. The fading, path loss, and $N_0$ are in a general configuration, and the cell radius and power depend on the experience of operator. The setting of SIS is based on the current level of self-interference suppression technology, designed to be an easy-to-achieve value.

We assume that one CU is assigned with one subcarrier, and the transmit power of the BS is uniformly distributed in frequency; hence, the transmit power from the BS to CUj is $P_{B,j}=P_{BS}^{\max }/L$. It is worth mentioning that most of the weight of the three-tier SIS architecture comes from the components required to cancel nonlinear SI in the RF chain. Considering the payload limitations of UAVs, for the communication transceivers mounted on UAVs, we only consider using chips for baseband interference cancellation and employing antenna isolation and air interface SIS techniques. Spatial SI can achieve 50–60 dB suppression through simple antenna isolation techniques and spatial self-interference cancellation algorithms [17,18], and the base band can use deep learning chips to predict and reconstruct the transmitted signal for interference cancellation, achieving 40–50 dB cancellation depending on the chip’s computational capability [29]. Therefore, we chose to examine the simulation results with a SIS capability of 110 dB.

Two metrics are used to evaluate the performance of the scheme; one is spectral efficiency, i.e., the sum rate, of the cell, and the other is D2D’s access rate, which is defined as the ratio of accessed DUs to the total DUs. To verify the superiority of the FD-MaxSumCell-HA scheme, we compare it with traditional half-duplex D2D underlaying networks. Moreover, we consider three scenarios of cellular users for each implementation: (1) There are only uplink CUs in the cell. (2) There are only downlink CUs in the cell. (3) There are joint uplink CUs and downlink CUs (JUAD) in the cell. This is carried out to eliminate the randomness of the user’s transmit direction

As shown in Figure 3a, whether using FD or traditional HD, D2D underlaying cellular networks can greatly increase the sum rate of the cell compared with networks that only have CUs, and the FD-MaxSumCell-HA scheme further improves the sum rate compared with the HD-D2D scheme. In particular, when there are 20 DUs in the cell, the sum rate of FD-MaxSumCell-HA shows a notable improvement of 43% compared to the HD-D2D scheme. Specifically, in the JUAD scenario, the sum rate of FD-MaxSumCell-HA is 1129.86 bps/Hz, surpassing the conventional HD scheme, which achieves a sum rate of 792.48 bps/Hz. The reason is that the sum rate of DUs improved nearly twofold because of the co-frequency co-time full-duplex adopted in D2D. Although, due to the residual SI and other interference introduced by dual-direction transmission, the improvement never reached twofold, the performance of the overall cell improved greatly. As for the access rate depicted in Figure 3b, as the number of DUs increases, the access rate of DUs for the two schemes monotonically decreases. This is because the more reuse occurs in the same subcarrier, the more interference is introduced, which will cause the DUs to not satisfy the requested QoS and exhibit access failure. However, the FD-MaxSumCell-HA scheme decreases slowly compared with the traditional HD-D2D scheme. This is because FD improves the spectral efficiency of D2D and makes it easier for DU to meet the QoS requirement. Therefore, both metrics are improved when the system adopts the FD-MaxSumCell-HA scheme.

To verify the effectiveness of HA proposed in this paper, we set the scheme in which CUs and DUs are randomly paired as the benchmark for comparison. And we only consider the JUAD scenario in this comparison. As illustrated in Figure 4a, both FD-D2D adopting HA and HD-D2D adopting HA perform better than them adopting random pairing in the sum rate comparison. This is because there is more severe interference when the CU is close to the DU in the same pair, and random pairing increases the chance of this. In particular, HD-D2D adopting HA is even better than FD-D2D with random pairing. This means that the gain brought by an excellent pairing algorithm is superior to the enhancement of the duplex mode. Figure 4b depicts the access rate comparison, where the HA scheme remains superior to the random pairing scheme. The access rate of FD and HD adopting the random scheme is even less than 50%.

Figure 5 shows the system performance comparison between the FD-MaxSumCell-HA and HD-D2D underlaying networks and the SIS of FD-D2D. The sum rate of the FD-MaxSumCell-HA scheme monotonically increase as SIS increases. FD-MaxSumCell-HA performs better even when SIS is low, which is easy to implement via antenna isolation. The access rate of FD-MaxSumCell-HA remains superior to the HD-D2D scheme when SIS is from 70 dB to 125 dB. However, when the access rate reaches 95%, it no longer improves with an increase in SIS. This is because mutual interference incurred by CU and DU reuse replaces RSI as a main factor, which depends on the resource coordination scheme.

Figure 3, Figure 4 and Figure 5 illustrate that our designed algorithm is highly robust. Compared to random allocation and traditional HD transmission, the combination of FD and the HA algorithm provides significant performance gains for the proposed network model. Therefore, even if the CSI reporting period is too long, causing some channel gain estimates to be inaccurate, the proposed model and algorithm can still enhance the cell’s spectral efficiency in most cases.

The reason for the performance enhancement of the FD-D2D underlaying network is that FD-D2D devices can improve the SE of the cell nearly twofold compared to traditional HD-D2D devices. As can be seen in Figure 6, in our scheme simulation, the sum rate of the FD-D2D pairs is approximately 1.73 times than that of HD-D2D pairs when DUs full load. This phenomenon leads to the BS being more inclined to allocate resource to DUs. The proposed scheme tends to be unfair for CUs, which results in more severe degradation of the performance of CUs. As demonstrated in Figure 7, the SE of CUs in the FD-MaxSumCell-HA scheme declines more sharply than that in the HD-MaxSumCell-HA scheme, and it is only 60% of the HD-MaxSumCell-HA scheme when DUs are equal to 20 in the joint uplink and downlink user scenario. But from another perspective, the system may seek to shift more traffic from CUs to DUs in certain scenarios; hence, this phenomenon is not always detrimental to wireless systems.

5. Conclusions

In this paper, which aims to further improve spectral efficiency, flexibility, and speed, we propose a novel FD-UAV-aided D2D network model and develop an FD-MaxSumCell-HA scheme, which adopts FD technology in UAV linking, to maximize the sum rate of the overall system. The optimization problem is MINLP, which is NP-hard and mathematically intractable. Thus, we decompose the problem into two subproblems, i.e., power allocation and user pairing, to solve it. The numerical results demonstrate that our proposed FD-MaxSumCell-HA scheme is superior to traditional HD-D2D underlaying cellular networks in both the system sum rate and access rate of D2D. In particular, when there are 20 CUs and 20 DUs in the cell, the sum rate of the FD-MaxSumCell-HA scheme improves by 43% against the traditional scheme. Moreover, the proposed scheme is better than traditional ones even when SIS is only 70 dB, which is easy to implement. Therefore, FD-MaxSumCell-HA has good application prospects in actual networks. However, in this paper, we do not consider the channel uncertainty caused by UAV mobility and perturbation, which is a problem to be solved in future research.