Miniature UAV-Aided Cooperative THz Networks with Reconfigurable Energy Harvesting Holographic Surfaces

1 Miniature UAV Empowered Reconfigurable Energy Harvesting Holographic Surfaces in THz Cooperative Networks Yifei Song, Jalal Jalali, Member, IEEE, Yanyu Qin, Filip Lemic, Member, IEEE, Jeroen Famaey, Senior Member, IEEE, and Natasha Devroye, Fellow, IEEE Abstract—This paper focuses on enhancing the energy efficiency (EE) of a cooperative network that features a miniature unmanned aerial vehicle (UAV) operating at terahertz (THz) frequencies and equipped with holographic surfaces to improve network performance. Unlike traditional reconfigurable intelligent surfaces (RIS), which serve as passive relays for signal reflection, this work introduces a novel concept: energy harvesting (EH) using reconfigurable holographic surfaces (RHS). These surfaces provide more powerful and focused energy delivery during wireless power transfer than RIS and are mounted on the miniature UAV. In this system, a source node enables the UAV to simultaneously receive both information and energy signals, with the harvested energy powering data transmission to a specific destination. The EE optimization problem involves adjusting nonorthogonal multiple access (NOMA) power coefficients and the UAV’s flight path while accounting for the unique characteristics of the THz channel. The problem is solved in two stages to maximize EE and meet a target transmission rate. First, the UAV trajectory is optimized using a successive convex approximation (SCA) method, followed by the adjustment of NOMA power coefficients through a quadratic transform technique. Simulation results demonstrate the effectiveness of the proposed algorithm, showing significant improvements over baseline methods. Index Terms—Cooperative communication, energy efficiency (EE), energy harvesting (EH), reconfigurable holographic surfaces (RHS), and miniature unmanned aerial vehicles (UAV). I. INTRODUCTION BUILDING on the original vision of 5G and extending into 6G, future wireless networks are expected to support enhanced mobile broadband (eMBB), ultra-reliable and low latency communications (URLLC) massive machine types communications (mMTC) with the requirements of improving data rates, improving network capacity, reducing latency, and minimizing energy consumption [1]. Researchers are investigating new network topologies focusing on establishing extensive backhaul links to boost network capacity [2]. Unmanned aerial vehicles (UAVs) have gained significant attention due to their unique advantage of establishing line-of-sight (LOS) Yifei Song and Yanyu Qin are with the Department of Electrical and Computer Engineering and Computer Science, Virginia Tech, USA, respectively. Jalal Jalali and Jeroen Famaey are with IDLab research group, University of Antwerp - imec, 2000 Antwerp, Belgium. Filip Lemic is with AI-Driven Systems Lab, i2Cat Foundation, Spain, and Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia. Natasha Devroye is with the Department of Electrical and Computer Engineering, University of Illinois Chicago, Chicago, IL, USA. The corresponding author is Yifei Song ([email protected]). This work was supported by the CHIST-ERA grant (CHIST-ERA-20-SICT003), with FWO (V478223N), ANR, NKFIH, and UKRI funding, European Union’s Horizon Europe programme Grant (101139161 — INSTINCT and 101192521 - MultiX projects), and in part, by the U.S. National Science Foundation under Grant CNS-2225511. communication links, enabling them to provide services to ground users while meeting quality of service (QoS) requirements. Specifically, miniature UAVs [3], including nanoUAVs—characterized by palm-sized dimensions and weights below 250 grams—and micro-UAVs, which are backpackportable and weigh between 250 grams and 20 kilograms, are particularly well-suited for operations in confined spaces due to their agility and compact design [4]. In contrast to standard UAVs, which may exceed 150 kilograms, miniature UAVs can safely operate in close proximity to humans [3], making them ideal for indoor applications [5]. These capabilities enable advanced use cases, such as deployment in industrial indoor rich-scattering environments [6], precise environmental monitoring [7], emergency search-and-rescue in collapsed structures [8], and enhanced immersive virtual reality [9] experiences through real-time mapping and interaction. Simultaneously, ensuring massive connectivity within the mMTC framework remains a significant challenge, as accommodating trillions of devices within the already congested and limited sub-6 GHz spectrum becomes increasingly difficult. Moreover, the rapid growth of the Internet of Things (IoT), with millions to billions of ubiquitously connected devices, presents even more severe challenges in supporting such largescale connectivity. To address these limitations, a shift towards higher-frequency terahertz (THz) communication is being actively explored, offering the potential for data rates in the hundreds of gigabits per second.Additionally, non-orthogonal multiple access (NOMA) [2], [10] is gaining traction as a method to support multiple users simultaneously on the same frequency and time slots, using efficient interference cancellation techniques. Together, these advancements aim to meet the growing demands of next-generation communication networks. More recently, the integration of UAVs with reconfigurable intelligent surfaces (RIS) has emerged as a promising approach to enhance wireless communication by utilizing the high mobility of UAVs and the ability of RIS to control signal reflections [21]. RIS technology, typically used as passive relays, adjusts the phase of incoming signals to improve coverage and capacity without requiring active amplification. However, this paper builds on this concept by considering Energy Harvesting (EH) using Reconfigurable Holographic Surfaces (RHS) [20], introducing a new aspect to RIS. We are the first propose the use of RHS to not only control surface reflections, but also for EH using wireless power transfer. RHS offers unique advantages for UAV integration due to its compact design and low power consumption [22]. RHS can be programmed to manipulate impinging electromagnetic 2 TABLE I: Overview of RISand RHS-assisted UAV works with focus on EH, objectives, and optimization approaches. Ref. EH RHS Objectives Optimization Parameters Algorithms [11]# # Enhance EE by optimizing UAV trajectory and number of RIS elements UAV weight, RIS weight, coverage probability ALOHA & Code Combining MAC protocol [12]# # Enhance EE via UAV power allocation and RIS phase shift optimization Power allocation, phase-shift matrix DRL [13]! # Enhance EE using time-space RIS-assisted EH scheme QoS, trajectory, resource allocation DRL [14]! # Maximize harvested energy using RIS-assisted EH while maintaining QoS Passive reflect-arrays, resource allocation DRL [15]! # Maximize bidirectional throughput in RIS-assisted UAV networks Time allocation, transmit power, EH ratio, trajectory Block coordinate descent & unsupervised learning [16]! # Enhance end-to-end throughput in UAV-enabled EH relay network Throughput, transmit power, trajectory Block coordinate descent & Lagrange duality [17]! # Evaluate coverage with non-linear EH model Coverage probability, trajectory, transmission power, harvesting energy Stochastic geometry [18]# ! Maximize sum rate in RHS-assisted UAV communication UAV position, digital beamforming, holographic beamforming weights Gradient-ascent [19]# ! Optimize the sum rate for multi-user communications Digital & holographic beamforming, receive combining Coordinate ascent [20]# ! Maximize the sum rate using holographic-assisted beamforming for LEO satellite communications Digital & holographic beamforming Dynamic programming This work ! ! Enhance EE and transmission rate in holographic EH UAV networks Power allocation parameters, UAV trajectory SCA (EM) fields in a highly customized manner. Their integrated circuitry and embedded sensors can be powered through energy harvesting modules, eliminating the need for dedicated power sources, power amplifiers, RF chains, or complex signal processing units [23]. This energy-neutral operation makes RHS highly attractive from an energy efficiency standpoint. Furthermore, RHS can be seamlessly integrated into wireless communication environments, offering advantages such as low power consumption, reduced hardware costs, and lightweight form factors. These features make them suitable for deployment in various settings, including warehouses, rooms, factory ceilings, and even human wearable applications [24]. Moreover, since UAVs are typically constrained by limited battery capacity and must manage both communication and movement, energy efficiency (EE) becomes a critical concern. In this context, RHS significantly enhances the EE of UAV networks by harvesting energy from the impinging signals, thereby reducing the reliance on external power sources. This EH capability, combined with optimizing flight paths and improving communication quality of service (QoS), is crucial for maximizing the overall EE of THz UAV-assisted networks empowered by RHS. The key contributions are as follows: ‚To the best of our knowledge, this is the first work to explicitly integrate RHS for EH within a novel THz NOMA cooperative communication model. In this model, the source node transmits: (1) a superimposed message to both the UAV and the destination node using distinct power allocation coefficients, as per NOMA principles; and (2) a dedicated power signal to the reconfigurable holographic surface (RHS) for EH, thereby extending its operational lifetime. ‚An optimization problem is formulated and solved to optimize the NOMA power allocation coefficients and the UAV trajectory in a three-dimensional (3D) system, aiming to maximize both the system’s EE and the target transmission rate. The paper is organized as follows: Section II introduces the related work. Section III presents the system model and optimization problem. Section IV covers the solution approach. Section Vdiscusses simulation results, and Section VI concludes with future research directions. In this paper, the notation is as follows: non-bold lowercase letters adenote scalars, bold lowercase letters arepresent vectors, bold uppercase letters Arepresent matrices, and calligraphic letters Adenote tensors. The symbol p¨qTindicates the transpose operation. The set of real numbers is represented by R. For the convenience of readers, the notations used throughout this paper are summarized in Table II. II. RELATED WORK Some studies have proposed energy-aware UAV-RIS models to enhance the number of tasks completed per flight [11]. For example, Nguyen et al. [12] formulate an EE maximization problem and employ deep reinforcement learning to jointly optimize UAV power allocation and the RIS phase shift matrix. Similarly, Kumar et al. [13] and Peng et al. [25] address power allocation with EH in a RIS-assisted UAV network under a dynamic wireless environment, using a deep reinforcement learning framework to enhance energy efficiency. Xiao et al. [26] introduced a solar-powered UAV-mounted RIS that provides external propulsion power and maximizes EE by optimizing the UAV trajectory alongside the beamforming active states. Tyrovolas et al. [27] studied a harvest-and-reflect (HaR) protocol designed to harvest energy for information transmission. Lyu et al. [28] explored a hybrid access point that transfers energy to both the RIS and users, enabling self-sustainable information transmission following the EH process. There are also inherent challenges when considering the use of THz for EH in UAV-RHS systems. From a materials science perspective [29], THz waves are readily absorbed by materials, particularly biological substances that resonate at THz frequencies. This effect is especially significant in polar molecules, e.g., water, where THz radiation induces dipole moments, enhancing absorption. Consequently, the high 3 Y Z Relay UAV X Destination node hsu[n] gsr[n,m] Source node Information signal Power signal UAV trajectory hsd[n] u[n] (a) Episode 1: Direct Transmission and EH with RHS. Source node Relay UAV Y Z X Re-transmit information signal UAV trajectory Destination node u[n] (b) Episode 2: Cooperative Transmission. Fig. 1: Illustration of cooperative transmission by miniature UAVs in a THz network empowered by EH RHS. (a) Episode 1: Direct information transmission from the source node to the destination node, information transmission from the source node to the relay UAV, and power signal transmission from the source node to the UAV. (b) Episode 2: Information re-transmission from the relay UAV to the destination node using the energy harvested via EH RHS. absorptivity of THz waves poses challenges for their use in wireless communication. We hypothesize that if RHS for EH at THz frequencies is developed in the future, it could be made from materials that leverage this absorptive property of THz waves to enhance EH efficiency. To our knowledge, this has not yet been done. Although there have been advancements [13]–[17] considering EH in UAV networks, none have focused on RHS for this purpose. Similarly, works [18]–[20] have considered RHS for various applications, but not for EH. Table Ipresents a comparative overview of recent UAV communication studies assisted by RIS and RHS. It outlines whether EH and RHS are incorporated, summarizes the main objectives, lists key optimization variables, and details the algorithms adopted in each work. This overview helps contextualize the unique contributions and methodologies of the current work in relation to the existing literature. In this paper, we harness the absorptive properties of the THz spectrum by equipping a miniature UAV with RHS for EH. This strategy extends battery life during data transmission and introduces a novel cooperative communication framework for air-to-ground transmission. III. SYSTEM MODEL AND PROBLEM FORMULATION We study a downlink NOMA transmission scenario within a miniature UAV-assisted RHS cooperative framework. The cooperative communication unfolds over two episodes. In the first episode, the source node employs NOMA to simultaneously transmit to both the destination and the miniature UAV. During this episode, the UAV performs EH via the RHS while decoding the source’s information, and the destination directly receives its data. In the second episode, the UAV acts as an aerial relay, forwarding the decoded data to the destination using the energy harvested in the first episode. In Fig. 1, the source node communicates with two terminals: a miniature UAV and a receiver destination node. The UAV serves as an EH-RHS to guarantee the high rate requirement of the destination node. A 3D coordinate system is utilized, where the source and destination are positioned at sptq “ rsxptq, syptq, HssTPR3ˆ1and dptq “ rdxptq, dyptq, HdsTP R3ˆ1, respectively. The destination node remains static on the ground, while the altitude of UAV with RHS and the source maintain fixed altitudes, though they differ from one another, i.e., Hu“Hr(altitude of RHS) and Hs. At any given time 0ătăT, the UAV’s instantaneous position is represented as uptq“rxptq, yptq, HusTPR3ˆ1. Moreover, the coordinates of the RHS-equipped UAV are expressed by rpt, mq“rxpt, mq, ypt, mq, HusTPR3ˆ1, where m“ t1, . . . , Murefers to the index of each holographic element. The total flight time of the UAV, denoted as T, is divided into Nequal time slots, with the trajectory at each time slot denoted as urns,@nP t1, ..., Nu. Each slot is small enough to treat the UAV position as nearly constant. The UAV’s position and speed are subject to the following constraints: ur1s “ us,(1a) urN`1s “ ue,(1b) }urn`1s´urns} ď ∆tVmax,@n, (1c) where Vmax is the maximum allowable speed, ∆tdenotes the length of each time slot, and usand uerepresent the UAV’s starting and ending positions, respectively. The channel coefficients for source-to-UAV and UAV-to-destination are hsurnsand hudrns, which adhere to the free-space path loss model and are expressed as: hsurns “ ı0 }urns´srns}e´ξpfq 2}urns´srns},@n, (2) hudrns “ ı0 }urns´drns}e´ξpfq 2}urns´drns},@n, (3) and the channel gain between the source-to-RHS is given by: gsrrn, ms “ ı0 }rrn, ms´srns}e´ξpfq 2}rrn,ms´srns},@n, m. (4) The THz path loss is represented by the exponential term, where ξpfqis the molecular absorption coefficient, influenced by frequency fand water vapor concentration [30]. For simplicity, we denote it as ξ, fixing the f. The reference power 4 TABLE II: Summary of Notation Symbol Description urnsUAV’s 3D position at time slot n srnsSource node position at time slot n drnsDestination node position at time slot n rrn, msPosition of m-th RHS element at slot n TTotal UAV operation time NNumber of time slots Vmax Maximum UAV speed ∆tDuration of each time slot hsurns,hudrns,hsdrnsChannel gains from source-to-UAV, UAVto-destination, and source-to-destination gsr rn, msChannel gain from source to m-th RHS element ξpfqMolecular absorption coefficient ı0Reference path gain srnsTransmitted NOMA signal at time slot n s1rns,s2rnsInformation symbols intended for UAV and destination, respectively ”1rns,”2rnsPower allocation coefficients for NOMA signals ppeak,pmax Max instantaneous and average source transmit powers ς1rns,ς2rnsPower split ratios for ID and EH antennas ˙ ˘ arms,˙ ˘ aMAbsorption coefficient of m-th RHS element and uniform coefficient ωrms,ωMPhase shift of m-th RHS element and uniform phase shift fp1q dÐurns,fp1q rrnsSINR at UAV decoding destination’s and own signals in Episode 1 fp1q drns,fp2q drns,fMRC drnsSINR at destination in Episodes 1, 2, and combined by MRC ErnsEnergy harvested by RHS at time slot n prns,pEH Minimum required harvested power and harvested power at EH RHS ηEnergy harvesting efficiency TrnsTransmission fraction of Episode 1 at time slot n pRHSrnsUAV transmit power in Episode 2 powered by harvested energy pcUAV circuit power consumption psumrnsTotal system power consumption at time slot n RsumrnsSum rate at time slot n ηEErnsEnergy efficiency at time slot n fminrns,γminrnsMinimum SINR thresholds at UAV and destination ϱ1p1qrns,ϱ2p2qrnsAdditive noise at destination during Episodes 1 and 2 ε2 1rns,ε2 2rnsNoise power at destination during Episodes 1 and 2 α,βAuxiliary parameter vectors for fractional objective transformation gain ı0“c{4πf, with cas the speed of light [30]. The channel power gain hsdrnsbetween the source and destination follows a similar structure as in Eq. 2and Eq. 3[31]. A. Episode One: Direct Transmission and EH with RHS In this episode, the source sends information to both the miniature UAV and the destination node using power-domain NOMA. The UAV, equipped with an RHS, acts as an EH user in this episode. The radio frequency (RF) source transmitted signal is: srns “ a”1rnss1rns`a”2rnss2rns,@n, (5) where s1rnsand s2rnsrepresent the symbols transmitted in each time slot, modeled as independent circularly symmetric complex Gaussian (CSCG) variables with zero mean and unit variance. Furthermore, a”1rnsand a”2rnscorrespond to the power allocation coefficients for NOMA in the n-th time slot, subject to the following constraints: ”1rns`”2rns ď ppeak,@n, (6a) 1 N N ÿ n“1 ”1rns`”2rns ď pmax,(6b) where ppeak is the maximum power the source can transmit in any time slot, and pmax is the total power constraint across all time slots. The signal received by the information decoding (ID) antenna and the absorptive EH RHS elements on the miniature UAV from the RF source can be expressed as: yp1q ID rns “ aς1rnshsurnssrns`zp1q 1rns,@n, (7) yp1q EH rns “ aς2rns M ÿ m“1 gsrrn, ms˙ ˘ armsejωrmssrns `zp1q 2rns,@n, (8) where zp1q 1rns „ Np0, ϵ2 1qand zp1q 2rns „ Np0, ϵ2 2qrepresent the CSCG noise at the UAV’s ID antenna and the EH RHS, respectively. Besides, 0ăς1rns, ς2rns ă 1are the received ID and EH power factors. The parameters ˙ ˘ armsand ωrmsdenote the absorption coefficient and phase shift applied by the m-th element of the RHS. Remark 1: The absorption coefficients on the RHS are assumed to be uniform, i.e., ˙ ˘ arms “ ˙ ˘ aM,@m. Nonlinearity and hardware impairments are not considered in this analysis. Additionally, no phase shift optimization is performed at the RHS, with the phase shifts uniformly set as ωrms “ ωM,@m. The miniature UAV utilizes successive interference cancellation (SIC) to decode the incoming signals. Specifically, it first decodes the destination node’s data and then subtracts it from the received signal to retrieve its own data. The signal-tointerference-plus-noise ratio (SINR) at the UAV for detecting s2rnsis given by: fp1q dÐurns “ ”2rns|hsurns|2 ”1rns|hsurns|2`ϵ2 1rns{ς1rns,@n. (9) Next, the SINR for decoding the miniature UAV’s own data is expressed as: fp1q rrns “ ς1rns”1rns|hsurns|2 ϵ2 1rns,@n. (10) Based on Eq. 7and Eq. 8, the RF power harvested by the EH RHS of the miniature UAV, neglecting the noise power, can be expressed as [27]: Erns “ η˙ ˘ aMe2jωMTrnsς2rnsˇˇˇˇˇ M ÿ m“1 gsrrn, msˇˇˇˇˇ 2 ,@n, (11) where ηP p0,1sis the energy conversion efficiency, and Trnsrepresents the transmission time fraction for the first episode within the n-th time slot, assuming equal transmission durations for both episodes. Therefore, the UAV’s transmit power in the second episode, empowered by the EH RHS, can be written as: pRHSrns “ Erns 1´Trns,@n. (12) 5 The received signal at the destination is given by: yp1q drns “ hsdrnssrns`ϱ1p1qrns,@n, (13) where ϱ1p1qrns „ Np0, ε2 1rnsq is the received noise at the destination node during the first episode. The SINR at the destination then becomes: fp1q drns “ ”2rns |hsdrns|2 ”1rns |hsdrns|2`ε2 1rns,@n. (14) B. Episode Two: Cooperative Transmission In this episode, the UAV utilizes the power harvested by the RHS, Eq. 12, to relay the destination node’s data. Consequently, the signal received at the destination node is: yp2q drns “ apRHSrnshudrnss2rns`ϱ2p2qrns,@n, (15) where ϱ2p2qrns „ Np0, ε2 2rnsq is the noise at the destination node. The corresponding SINR is given by: fp2q drns “ η˙ ˘ a2 Me2jωMς2rnsˇˇˇˇ M ř m“1 gsrrn, msˇˇˇˇ 2 |hudrns|2 ε2 2rns,@n. (16) Finally, the destination node applies maximal ratio combining (MRC) to integrate the signals received in both episodes. The overall SINR can be expressed as: fMRC drns “ fp1q drns`fp2q drns,@n. (17) C. Resource Allocation Problem Formulation We begin by defining the network’s EE as the ratio of the total sum rate to the total power consumed by the network. Mathematically, this is represented as ηEErns “ Rsumrns psumrns, where Rsumrns “ log2p1`f1 rrnsq ` log2p1`fMRC drnsq. Assuming a constant power consumption for the miniature UAV’s flight, pc, the total transmission power of the system can be written as: psumrns “ ”1rns`”2rns`pc´pRHSrns. To maximize the EE by optimizing the NOMA power allocation coefficients and the UAV’s trajectory, we formulate the following optimization problem: P1: max ,”1rns,”2rns,urnsÿN n“1ηEE rns(18) s.t. :1 NÿN n“1pRHSrns ě 1 NÿN n“1prns,(18a) fp1q dÐurns ě fminrns,@n, (18b) fMRC drns ě γminrns,@n, (18c) prns ě 0,@n, (18d) (1a)´(1c),(6a),(6b). The constraint Eq. 18a ensures that the power harvested by the EH RHS of the miniature UAV over all time slots is greater than the minimum required harvested power prns “ Erns Trns (where prns “ pEH, representing the harvested power). Constraint Eq. 18b guarantees successful decoding of the destination node’s data at the UAV, with the SINR exceeding the threshold fminrns, while Eq. 18c enforces that the destination node’s SINR remains above the minimum requirement γminrns, where γmin ěfmin. Finally, Eq. 18d ensures that the UAV’s transmitted power is feasible and non-negative. IV. A TWO-STEP SEQUENTIAL APPROACH TO SOLVING THE EE OPTIMIZATION PROBLEM The optimization problem P1is NP-hard and non-convex due to the interdependence among the optimization variables. Additionally, the objective function in P1is a sum of ratios, which makes traditional Dinkelbach method approaches unsuitable [32]. To address this, we propose a two-step approach that separates the optimization process, allowing each variable to be optimized independently. A. Step One: Optimizing EH RHS Miniature UAV Trajectory In this step, the trajectory of the miniature UAV is optimized while the NOMA power allocation coefficients remain fixed. The sum rate function remains non-convex due to the coupling of the optimization variables. However, to address this, the non-linear fractional objective function is first transformed into a subtractive form [33]. Theorem [33]: Let u˚rnsbe the optimal solution to P1. Then, given the existence of two vectors, α“ rα˚ 1, . . . , α˚ NsT and β“ rβ˚ 1, . . . , β˚ NsT, the following optimization problem provides an optimal solution as follows: P2: max urnsÿN n“1α˚ n“Rsumrns´β˚ nppsumrnsq‰.(19) Moreover, u˚rnsmust satisfy the following conditions: R˚ sumrns´β˚ nppsumrnsq “ 0,@n, (20) 1´α˚ nppsumrnsq “ 0,@n. (21) The equivalent subtractive form in Eq. 19, using the additional parameters α˚,β˚, shares the same optimal solution as P1 for fixed values of ”1rnsand ”2rns. Specifically, Eq. 19 can be solved iteratively using a two-layer approach consisting of inner and outer layers. In the inner layer, Eq. 19 is solved with fixed values of αand β. Then, Eq. 20 and Eq.21 are updated in the outer layer to find the optimal tα˚,β˚u. 1) Inner-layer:Here, we optimize the trajectory based on the optimal NOMA power allocation coefficients as follows P3: max urnsÿN n“1α˚ n“Rsumrns´β˚ nppsumrnsq‰(22) s.t. : N ÿ n“1 c1e´ξp}urns´srns}q }urns´srns}2ě N ÿ n“1 prns,(22a) ”2rns ”1rns`c2}urns´srns}2eξp}urns´srns}q ěfminrns,@n, (22b) c1ı2 0 ε2 2rns¨e´ξp}urns´srns}`}urns´drns}q }urns´srns}2}urns´drns}2(22c) `”2rns |hsdrns|2 ”1rns |hsdrns|2`ε2 1rnsěγminrns,@n, (1a)´(1c),(18d), where c1“ηM ˙ ˘ a2 Me2jωMς2rnsı2 0and c2“ϵ2 1rns ς1rnsı2 0 . The optimization problem P3remains non-convex. Therefore, P3 6 is reformulated into an equivalent form by introducing slack optimization variables, parns, brns, crns, drnsq, as follows: P4: max urns,arns,brns,crns,drnsÿN n“1α˚ n“Rsumrns´β˚ nppsumrnsq‰ (23) s.t. :ÿN n“1 c2 ecrnsěÿN n“1prns,(23a) ”2rns ”1rns`c2ecrnsěfminrns,@n, (23b) ”2rns |hsdrns|2 ”1rns |hsdrns|2`ε2 1rns`. . . `c1ı2 0 ε2 2rnsecrns`drnsěγminrns,@n, (23c) arns ď }urns´srns}2 e´ξ}urns´srns|,@n, (23d) brns ď }urns´drns}2 e´ξ}urns´drns},@n, (23e) arns ď ecrns,@n, (23f) brns ď edrns,@n, (23g) (1a)´(1c),(18d), where Rsumrns “ log2ˆ1`c2”1rns ecrns˙(24) `log2ˆ1`fp1q drns`ˆc1ı2 0 ε2 2rns¨1 ecrns`drns˙˙. Using these transformations, the main objective function and constraints become convex but still intractable. Therefore, successive convex approximation (SCA) using first-order Taylor expansions is applied to approximate P4as convex functions. The first-order lower bounds are given by: ecrnsěecpkqrnsp1`crns´cpkqrnsq ∆ “˜ecrns,@n, (25) edrnsěedpkqrnsp1`drns´dpkqrnsq ∆ “˜edrns,@n, (26) }urns´srns}2 e´ξ}urns´srns}ě››upkqrns´srns›› 2 e´ξ}upkqrns´srns} `p2`ξ||upkqrns´srns||q¨ (27) pupkqrns´srnsqTpurns´upkqrnsq e´ξ}upkqrns´srns} ∆ “}˜ urns´srns}2 e´ξ}˜ urns´srns},@n, (28) }urns´drns}2 e´ξ}urns´drns}2ě››upkqrns´drns›› 2 e´ξ}upkqrns´drns} `p2`ξ||upkqrns´drns||q¨ (29) pupkqrns´drnsqTpurns´upkqrnsq e´ξ}upkqrns´drns} ∆ “}˜ urns´drns}2 e´ξ}˜ urns´drns}2,@n, (30) where ecpkqrnsand edpkqrnsrepresent the Taylor expansion points at iteration k. With this transformation, P4’s approximation becomes: P5: max urns,arns,brns,crns,drnsÿN n“1α˚ n“˜ Rsumrns´β˚ nppsumrnsq‰ (31) s.t. :ÿN n“1 c1 ˜ecrnsěÿN n“1prns,(31a) ”2rns ”1rns`c2˜ecrnsěfminrns,@n(31b) ”2rns |hsdrns|2 ”1rns|hsdrns|2`ε2 1rns`c1ı2 0 ε2 2rns˜ecrns`drnsěγminrns,@n, (31c) arns ď }˜ urns´srns}2 e´ξ}˜ urns´srns},@n, (31d) brns ď }˜ urns´drns}2 e´ξ}˜ urns´drns},@n, (31e) arns ď ˜ecrns,@n, (31f) brns ď ˜edrns,@n, (31g) (1a)´(1c),(18d), where ˜ Rsumrns “ Rsumrns|ecrns“˜ecrns,edrns“˜edrns. Optimization solvers can be employed to find a solution for P5[32]. 2) Outer-layer:The damped Newton method is applied to find the optimal values for tα,βu. Let θnpβnq “ R˚ sumrns´ β˚ nppsumrnsqand θN`jpαjq “ 1´α˚ jppsumrjsq,jP t1, ..., Nu. As shown in [34], the solution tα˚,β˚uis optimal if and only if θpα,βq “ rθ1, θ2, ..., θ2NsT“0. The updated values of αi`1and βi`1can be computed by: αi`1“αi`ϑiµi N`1:2N,βi`1“βi`ϑiµi 1:N,(32) where µ“ r´ θpα,βqs´1θpα,βqwith ´ θpα,βqbeing the Jacobian matrix of θpα,βq, and ϑiis the largest value of Πmat iteration isatisfying: (33) }θpαi`Πmµi N`1:2N,βi`Πmµi 1:Nq} ď p1´℘Πmq}θpα,βq}, where mP t1,2, ...u,ΠmP p0,1q, and ℘P p0,1q. B. Step two: Optimizing NOMA Power Allocation Coefficients Consider the following sum-fraction optimization problem: min ΩPC J ÿ j“1 AjpΩq BjpΩq,(34) where Jrepresents the total number of fractional terms, and Ωis the vector of optimization variables within the feasible domain C. It can be shown Eq. 34 is equivalent to: min ΩPC,ϖją0 J ÿ j“1 ϖjA2 jpΩq` J ÿ j“1 1 4ϖj 1 B2 jpΩq.(35) The solution to both Eq. 34 and Eq. 35 is identical. It is worth noting that if BjpΩqis concave and AjpΩqis convex, then problem in Eq. 35 becomes a convex quadratic problem for the given ϖj. Building on this, the convex problem in Eq. 35 7 Lzp”1rns,”2rns,a,b,Υ,c,dq “ N ÿ n“1 ϖrnsp2 sumrns` 1 2z«ˆ„ N ÿ n“1 an`z´ϵ2 1rns ς1rns´”2rns|hsurns|2 fminrns`”1rns|hsurns|2¯ȷ`˙2 ` N ÿ n“1 1 4ϖrnsˆ R2 sumrns`ˆ„ N ÿ n“1 bn`zpε2 1rnsχrns´”2rns|hsdrns|2`”1rns|hsdrns|2χrnsqȷ`˙2 `ˆ„ N ÿ n“1 Υn`zp”1rns`”2rns´ppeakqȷ`˙2 `ˆ„cn`zp1 N N ÿ n“1 ”1rns`”2rns´pmaxqȷ`˙2 `ˆ„ N ÿ n“1 dn´zprnsȷ`˙2 ´ˆN ÿ n“1 a2 n`b2 n`Υ2 n`c2 n`d2 n˙ff,(42) is solved for a given ϖj“1{2BjpΩqAjpΩq, and the value of ϖjis updated in the next iteration. Thus, with a fixed UAV trajectory, problem P1can be rewritten in the following equivalent form: P6: min ”1rns,”2rns,ϖrnsą0 N ÿ n“1 ϖrnsp2 sumrns ` N ÿ n“1 1 4ϖrns 1 R2 sumrns(36) s.t. :”2rns|hsurns|2 fminrns´”1rns|hsurns|2ěϵ2 1rns ς1rns,@n, (36a) ”2rns|hsdrns|2´”1rns|hsdrns|2χrns ě ε2 1rnsχrns,@n, (36b) (6a),(6b),(18d), where χrns “ γminrns´ c1ˇˇˇˇ M ř m“1 gsrrn,msˇˇˇˇ 2 |hudrns|2 Mı2 0ε2 2rnsand ϖrns “ 1 2p2 sumrnsR2 sumrns. It is evident that all constraints are linear and convex. However, the objective function remains non-convex due to the non-concave nature of the sum rate function. To address this, we apply the result from the following corollary [32]. Corollary 1: Let Fbe a monotonically decreasing function of the ratio Cjp℧q Djp℧q. The optimization problem min ℧PC J ÿ j“1 FjˆCjp℧q Djp℧q˙,(37) is equivalent to: min ℧PC,λj J ÿ j“1 Fjˆ2λjbCjp℧q´λ2 jDjp℧q˙,(38) where λjis updated iteratively as: λj“?Cjp℧q Djp℧q. By applying the result from Corollary 1, the second term in the objective function of P6can be rewritten as: min ”1rns,”2rns,λrns N ÿ n“1 1 4ϖrns 1 ˆ R2 sumrns,(39) where ˆ Rsumrns “ log2p1`f1 rrnsq `log2ˆ1`f2 drns`2λrnsa”2rns |hsdrns|2 ´λ2rnsp”1rns |hsdrns|2`ε2 1rnsq˙,(40) with λrns “ ?”2rns |hsdrns|2 ”1rns |hsdrns|2`ε2 1rns. Here, ˆ Rsumrnsbecomes biconcave in terms of both the power allocation coefficients and λrns. Consequently, the multi-convex optimization problem is formulated as: P7: min גrns,ϖrns N ÿ n“1 ϖrnsp2 sumrns` N ÿ n“1 1 4ϖrns 1 ˆ R2 sumrns(41) s.t. :(6a),(6b),(18d),(36a),(36b), where גrns “ r”1rns,”2rnss P R2ˆ1. Note that psumrns depends on the power allocation coefficients, and each coefficient is subject to its respective constraints. Thus, psumrnsand ˆ Rsumrnsare decoupled to enable the distributed optimization of psumrns. To solve this, the augmented Lagrangian method (ALM) is applied, as defined in Eq. 42, where a penalty term is introduced in the Lagrange function of P7, yielding a sub-optimal solution. In Eq. 42,zrepresents the penalty factor, while a,b,Υ,c,dare the Lagrange multipliers. Finally, the complexity of the proposed solution is determined by solving P2, P5, and P7, with complexities of Op9N3q, Opp8N`3qp5Nq3q, and OpN2q, respectively. Thus, the overall complexity of the proposed two-step method is approximately polynomial of degree four [32]. V. SIMULATION RESULTS AND DISCUSSIONS Our simulation setup involves a scenario within a square area, each side being 30 meters, containing one user and a miniature UAV, both randomly placed. To minimize path loss peaks, the carrier frequency is set to f“1.2THz suggested by [35], [36] with a transmission bandwidth of 10 GHz. The model also considers the frequency-dependent absorption coefficient, ξpfq, which accounts for molecular absorption loss 8 TABLE III: Simulation Parameters for EE Maximization of THz-NOMA Networks Empowered by Holographic Surfaces for Miniature UAVs. Parameter Value Area side length 30 meters Carrier frequency 1.2THz Transmission bandwidth 10 GHz Absorption coefficient, ξpfq0.005 RHS absorption coefficient, ˙ ˘ aM1 Maximum miniature UAV flying speed, Vmax 1 meter{second Duration of each time slot, ∆t0.1 second Miniature UAV Operation time, T45 second Noise power spectral density ´174 dBm{Hz Source Node altitude, Hs2 meters Miniature UAV altitude, Hu3 meters Peak power, ppeak 1 W Circuit power, pc0.52 W 1 2 3 4 5 6 7 8 30 40 50 60 70 80 ¯psum [W] ηEE [Mbits{Joule] Proposed Solution Method A Method B Method C Method D Method E Initial Fig. 2: The impact of average network transmit power, ¯psum, on the EE of THzNOMA networks with a miniature UAV empowered by holographic surfaces due to water vapor [37]. All statistical results are derived from aggregating data gathered through an extensive set of simulation trials, which include 1000 random realizations of channel gains. This systematic approach provides an in-depth understanding of the dynamics associated with deploying and operating the miniature UAV under specified environmental conditions, offering essential insights for optimizing UAVassisted communication networks. A summary of all simulation parameters studied in this paper is presented in Table III, as suggested in [30], [38], [39]. To thoroughly assess the performance of our proposed resource allocation algorithm, we conducted a comparative study using the following benchmarks, each selected to highlight different system aspects: ‚Method A: Assesses the algorithm’s performance within a fixed NOMA framework, providing a baseline for how the algorithm performs with static power coefficients. ‚Method B: Compares NOMA and orthogonal multiple access (OMA) to identify which access scheme is more efficient, crucial for understanding the benefits of multiuser communication in this context. ‚Method C: Tests the algorithm with a fixed UAV flight path, isolating the effects of UAV trajectory optimization and measuring its contribution to overall performance. 5 10 15 20 25 30 35 40 20 25 30 35 40 45 50 Miniature UAV Operation time [s] ηEE [Mbits{Joule] Proposed Solution Method A Method B Method C Method D Initial Fig. 3: The EE versus the operational time of the miniature UAV-empowered holographic surfaces in the THz-enabled network. ‚Method D: Evaluates a scenario without RHS, focusing on consistent power splitting EH at the UAV antenna. This shows the impact of RHS on EH and EH. ‚Method E: Implements a fractional programming approach [40] without RHS for comparison, highlighting the benefits of incorporating RHS in our proposed solution. Fig. 2illustrates the EE dynamics as influenced by the average network transmission power, expressed as ¯psum “ pmax `ppeak `pc´pEH. In this figure, the ’Initial’ curve depicts the EE performance based on an initial, unoptimized (random) configuration of the miniature UAV’s flight path. A key finding from our analysis is that our proposed algorithm consistently surpasses various benchmark methods, with its advantage becoming more evident as ¯psum increases, resulting in a widening performance gap. The results demonstrated the effectiveness of our proposed approach, showing improvements of: 30.3% over Method E, 23.0% over Method D, 21.2% over Method C, 18.1% over Method B, and 7.26% over Method A. These results strongly affirm the proposed algorithm’s ability to significantly boost EE, proofing its effectiveness within miniature UAV-empowered RHS communication networks. Fig. 3offers a detailed examination of how the mission duration, represented by the miniature UAV’s operational time T, affects EE across various benchmark schemes. The analysis reveals an interesting pattern: as mission time increases, there’s a noticeable rise in EE for schemes utilizing fixed trajectories (Method D) and those starting with non-optimized but feasible configurations (‘Initial’). This improvement in EE is due to extended communication opportunities and the ability to adjust flight parameters over time. However, this trend is not consistent across all methods; specifically, Methods A, B, and C do not show the same EE increase as Tgrows. Quantitatively, extending the mission duration results in EE improvements of at least 37.1%,26.8%,22.8%,16.5%,and 12.8% when using Methods A–E, respectively. These gains indicate that longer mission times provide a strategic benefit by allowing the holographic surfaces-assisted miniature UAV to 9 optimize both communication metrics and flight adjustments, thereby enhancing the overall network QoS. Nevertheless, the relationship between mission time and EE is complex. The interplay among optimization variables produces a non-linear, though generally increasing, trend in EE as mission duration extends. This highlights the intricate dynamics of EE optimization, where certain adjustments can lead to substantial gains. The observation that mission duration significantly impacts EE emphasizes an important challenge: minimizing the task completion time for miniature UAVempowered holographic surfaces relay systems while meeting specific EE targets. This requires balancing operational efficiency and mission urgency, suggesting a rich area for further research into optimizing UAV-based communication networks, with or without EH capabilities. VI. CONCLUSION AND FUTURE WORK In this paper, we explored the complexities of improving the efficiency of a cooperative THz NOMA-based miniature UAV network powered by EH holographic surfaces. We began by formulating an EE optimization problem aimed at refining the network’s resource allocation strategy. A novel deployment plan for the miniature UAV was introduced, designed to enhance THz wireless connectivity while accounting for molecular absorption effects, a crucial element in the path loss channel gain model for THz-enabled UAVs. Building on this, we developed an optimization framework to enhance EE, ensuring it met stringent QoS requirements. The optimization targeted key decision variables, including miniature UAV positioning and NOMA power allocation coefficients, based on a two-episode iterative solution. 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