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Energy Minimization Oriented Resource Allocation for Integrated Sensing and Communication in Marine IoT Networks
arXiv:2607.13462v1 [cs.NI] 15 Jul 2026
Qianru Wang, Student Member, IEEE, Li Ping Qian, Senior Member, IEEE, Chenglong Dou, Haijun Zhang, Fellow, IEEE, and Yuan Wu, Senior Member, IEEE
Abstract—Integrated sensing and communication (ISAC) has become a promising technical framework for Marine Internet of Things (MIoT) systems. Nevertheless, all devices rely on battery power supply, so energy efficiency becomes a core bottleneck that limits practical deployment. This paper thus investigates the energy consumption minimization optimization problem of MIoToriented ISAC systems. In this system, an uncrewed aerial vehicle (UAV) employs non-orthogonal multiple access (NOMA) to simultaneously perform target sensing and collect data from uncrewed surface vehicles (USVs), and subsequently forwards both the processed sensing information and the data collected from the USV to a shore-based base station (SBS). Subject to latency limits and sensing performance requirements, the total system energy consumption can be minimized through joint optimization of multiple variables, including the UAV’s transmit beamforming, the dedicated sensing signal, the USVs’ transmit power, the UAV’s computation power, and the time resource allocation for sensing and communication stages. To address the established non-convex optimization problem, we construct a layered solution architecture, which splits the original problem into several separate subproblems and alternately optimizes each subproblem based on its own mathematical characteristics. In particular, we first derive the closed-form solution for the transmit power of USVs and then apply variable substitution. The successive convex approximation (SCA) technique is further adopted to transform the leftover nonconvex subproblems into convex forms, based on which we design corresponding efficient iterative solving algorithms. Numerical simulation outcomes verify the validity and accuracy of the proposed algorithm for total system energy reduction. Compared with the orthogonal frequency division multiple access (OFDMA) transmission scheme and the genetic algorithm, our algorithm reduces the system energy consumption by 19.71% and 8%, respectively. Moreover, the energy consumption optimized by our algorithm only deviates by 8.72% from the optimal value solved by the LINGO solver. Index Terms—Integrated Sensing and Communications, Marine Internet of Things, Energy Consumption Minimization, NonOrthogonal Multiple Access.
I. I NTRODUCTION The fast expansion of the Marine Internet of Things (MIoT) technologies greatly promotes practical scenarios including smart ocean monitoring, autonomous underwater vehicles, and maritime surveillance [1]–[3]. These applications require ultrareliable and energy-efficient wireless connectivity in harsh oceanic environments, where limited node energy, dynamic topology, and constrained bandwidth pose major challenges to network performance and longevity [4], [5]. Moreover, MIoT systems increasingly leverage integrated sensing and communications (ISAC) to acquire environmental data and
support intelligent maritime operations [6]–[8]. However, realizing low-latency data transmission and energy-efficient task execution across wide oceanic regions creates a prominent technical barrier, particularly when the system needs to satisfy simultaneous sensing and real-time communication requirements. Non-orthogonal multiple access (NOMA) serves as a robust technical support to realize massive device connections and spectrum-efficient communications in next-generation wireless networks [9]. By enabling multiple terminals to reuse identical time-frequency resource units via superposition coding (SC) and signal recovery with successive interference cancellation (SIC), NOMA significantly improves spectral efficiency while supporting differentiated user requirements [10], [11]. This feature makes NOMA particularly suitable for MIoT networks, where densely deployed sensor nodes frequently contend for limited spectrum resources. Notably, ISAC in IoT scenarios including MIoT often faces symbiotic interaction challenges between sensing and communication as well as security risks. These issues need to be addressed with core performance optimization, and integrating ISAC with 6G key technologies is an important direction to tackle these practical problems [12]. Moreover, NOMA effectively suppresses co-channel interference in ISAC systems, enabling reliable communication in dense user environments [13]. Applying NOMA to marine environments thus offers a promising approach to enhance both energy utilization and sensing accuracy under resource constraints. Integrating ISAC into MIoT systems offers a transformative solution to the spectral and energy constraints inherent in ocean environments [14], [15]. By combining sensing and communication capabilities under one unified architecture, ISAC enables more efficient spectrum utilization while supporting critical marine operations such as real-time ship detection, environmental monitoring, and underwater navigation [16]–[18]. This integration is particularly crucial in maritime scenarios, where conventional communication methods often suffer from severe signal attenuation and limited bandwidth. For ISAC-based MIoT networks, non-orthogonal transmission schemes such as NOMA and RSMA adapt well to resource-constrained marine environments while suppressing co-channel interference between sensing and communication signals [19]. However, the deployment of ISAC brings distinctive technical hurdles, including interference control difficulties and resource allocation balance conflicts between com-
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munication quality and sensing precision. Most existing ISAC efforts for MIoT either neglect the strict resource constraints of ocean environments or fail to jointly optimize communication and sensing with non-orthogonal schemes. To address these issues, this work leverages NOMA to jointly optimize the ISAC design, aiming to minimize energy consumption while meeting stringent latency and sensing performance requirements in dynamic marine networks. With the rapid expansion of marine monitoring and offshore industrial activities, a single-tier data collection framework relying solely on uncrewed surface vehicles (USVs) or static buoys is no longer sufficient to meet the growing demands of MIoT systems. Uncrewed aerial vehicle (UAV)-assisted ISAC networks, where UAVs serve as mobile relays for both data aggregation and real-time environmental sensing, enable more efficient utilization of communication and computing resources while overcoming the limitations of direct longdistance transmissions [20]–[22]. In this paper, we propose a NOMA-enabled ISAC framework for MIoT, where the UAV assists a group of USVs by enabling uplink communication via NOMA while concurrently performing radar sensing toward a target. Specifically, there are two phases. In Phase I, the UAV transmits a radar signal to probe the target, while concurrently collecting echo signals and uplink data from USVs via NOMA. In Phase II, the UAV relays the fused sensing and communication data to the shore-side base station (SBS). This integrated framework introduces two main challenges. First, in Phase I, the coexistence of sensing and communication results in inter-functional interference. Thus, it is critical to design an effective NOMA mechanism to mitigate such interference, enhancing communication performance without compromising sensing accuracy. Second, the allocation of time lengths for Phase I and Phase II brings an inherent tradeoff, and reasonable division of phase time can effectively cut down total system energy consumption while meeting the overall latency constraints. The key innovations of this work are summarized below: We propose an energy consumption minimization framework for ISAC in MIoT networks. In Phase I, the UAV simultaneously performs target sensing and collects data from a set of USVs using NOMA. The raw sensing data are processed into meaningful sensing information. In Phase II, the UAV transmits both the USVs’ data and sensing information to the SBS. To reduce the total energy consumption of UAV and USVs, we jointly optimize the transmit power of USVs, the transmit beamforming and computation power of the UAV, the dedicated sensing signal, and the phase time allocation, under the limits of latency and sensing rate requirements. • Considering the non-convex characteristic of the constructed optimization problem, we take advantage of its layered architecture and split the original model into two separate subproblems belonging to upper and lower layers. We further break down the lower-layer optimization task into three independent sub-tasks. Specifically, we first work out an explicit closed-form formula to calculate the transmit power assigned to each USV. Then, we •
propose a polyblock approximation-based algorithm to determine the UAV’s optimal computation power based on its monotonicity nature. Third, by identifying the rankone structure of the UAV’s transmit beamforming, we design an alternating optimization-based algorithm to obtain its solution. Finally, we further adopt successive convex approximation (SCA) to optimize the dedicated sensing signal and the durations in these two communication phases, respectively. • We carry out abundant numerical simulations to validate the performance and energy-saving capacity of the designed algorithm, and reveal its performance improvements against several mainstream benchmark strategies. The simulation data reveal that total system energy consumption can be cut by an average of 19.71% compared with the OFDMA benchmark scheme. The subsequent chapters of this paper are arranged in the following structure. Section II summarizes existing relevant research works. Section III illustrates the system setup and the formulated optimization problem. A high-performance solver built on decomposition theory is put forward in Section IV to find the optimal solution. Section V provides numerical simulation data to demonstrate the performance of the designed algorithm. At last, Section VI draws the whole paper to a close and puts forward possible directions for subsequent research. II. R ELATED W ORK In this section, we first review related studies on NOMAassisted communication and resource allocation. Then, we review studies on UAV-assisted data collection and ISAC. NOMA-assisted communication and resource allocation have gained significant research attention for enhancing system performance. In [23], Ting et al. constructed a collaborative optimization architecture targeting computation offloading and resource partitioning within UAV emergency communication systems supported by NOMA in order to cut computational overhead. In [24], Dai et al. introduced a hybrid semantic communication architecture for maritime networks to achieve energy minimization via joint optimization. In [25], Alishahi et al. put forward a collaborative resource distribution strategy for IRS-assisted FL-WPT systems integrated with NOMA to cut down system energy consumption. In [26], Li et al. explored mobility-adaptive task offloading and spectrum allocation for NOMA-supported MEC vehicular networks to shorten task processing delay. In [27], Xu et al. developed a unified resource scheduling strategy for NOMA-assisted IoT-MEC networks to reduce weighted cumulative energy consumption. In [28], Dou et al. devised a joint optimization framework for AP beamforming and two-tier resource allocation in NOMAassisted ISTTO systems to minimize total energy consumption. However, most existing works in NOMA-assisted communication focus on general scenarios and fail to consider the specific energy constraints and dynamic environment of MIoT. UAVs act as a core carrier to deploy integrated sensing and communication systems, and they take advantage of flexible mobile characteristics and the dual-function feature of ISAC to support simultaneous target sensing and high-efficiency communication in complex environments. In [29], Pang et al. put
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TABLE I C OMPARISON OF R ELATED W ORKS W ITH O UR W ORK Access Scheme
OMA
NOMA
Optimization Objective Achievable Rate Maximization Radar Estimation Rate Maximization Power Consumption Minimization Target Location Estimation Error Minimization Communication Throughput Maximization Computation Overhead Minimization Energy Consumption Minimization Energy Consumption Minimization Task Latency Minimization Energy Consumption Minimization Energy Consumption Minimization Computation Throughput Maximization Energy Consumption Minimization
Reference [29] [30] [31] [32] [33] [23] [24] [25] [26] [27] [28] [34] our work
forward a UAV-supported sensing-aided transmission strategy for vehicle networking scenarios built upon ISAC, which can dynamically tune sensing and communication performance. In [30], Liu et al. developed a UAV-aided ISAC architecture for IoT to boost radar estimation rate under communication rate constraints. In [31], Ata Khalili et al. explored collaborative resource scheduling and flight path planning for multi-terminal multi-target UAV-ISAC frameworks to cut overall power consumption. In [32], Pan et al. focused on a UAV-enabled ISAC system with OFDMA to minimize target location estimation error while ensuring communication QoS. In [33], Gu et al. studied UAV trajectory design for ISAC, addressing tradeoffs in different scenarios. In [34], Qin et al. built a NOMAsupported UAV ISCC network and adopted deep reinforcement learning to jointly adjust UAV flight trajectory, computing resource allocation, and beamforming vectors. However, existing UAV-assisted data collection and ISAC works neglect the high dynamics and uncertainty of the marine environment, energy limitations of UAVs, and the stringent latency and real-time requirements of sensing and communication in MIoT. Table I presents a comprehensive comparison among existing studies and our work across several key dimensions, including access schemes, optimization objectives, latency requirements, computation resource allocation, and ISAC integration in UAV-assisted networks. Although NOMA-based communication, resource optimization, UAV-assisted data collection, and ISAC have seen progress, challenges remain in MIoT networks. The marine environment’s high dynamics and energy limitations of UAVs, along with strict latency requirements, call for optimized transmission and computational resources. To tackle these problems, we construct an energy consumption minimization for the ISAC framework in MIoT networks. We adjust the UAV’s transmit beamforming, the dedicated sensing signal, the transmit power of all USV terminals, the UAV’s computation capacity, and phase time allocation together to cut total system energy expenditure and meet latency and sensing performance limits. III. S YSTEM MODEL AND PROBLEM FORMULATION We investigate the problem of energy consumption minimization for ISAC in MIoT networks. As shown in Fig. 1, the MIoT network is composed of multiple single-antenna USVs denoted as N = {1, 2, . . . , N }, a single-antenna SBS,
Latency Requirement X X X X X X X X X X X X X
Computation Resource Allocation X X X X X X X X X X X X X
ISAC in UAVs X X X X X X X X X X X X X
TABLE II NOTATIONS AND DEFINITIONS Symbols B UAV B BS cn C UAV Dn Dtarget USV En UAV E11 UAV E12 E UAV,hov E tot f UAV f UAV,max t2 gn HSI h n0 nBS pn pUAV,max pUSV,max n Q Qreq RUSV n RUAV s0 T1 T11 T12 T2 T tot,max T1max T2max u v w w UAV x0 xn e x0 y0 e yBS ω UAV δ τ Λ Γn
Descriptions The bandwidth from USV to the UAV The bandwidth from the UAV to the SBS The communication receive beamforming of the UAV The UAV’s required CPU cycles to process each single bit The volume of data transmitted by USV n The transmitted compressed sensing data The energy consumption of USV n The energy consumption of the UAV in T11 The energy consumption of the UAV in T12 The hovering energy consumption of the UAV The total energy consumption of the system The UAV’s CPU frequency The maximum computing power of the UAV The channel gain from USV n to UAV The residual self-interference channel at the UAV The channel gain between the UAV and the SBS The noise at the UAV The noise at the SBS The transmit power by USV n to the UAV The maximum transmit power of the UAV The maximum transmit power of the USV n The sensing estimation rate of the UAV The required sensing estimation rate The achievable transmission rate from USV n to the UAV The transmission rate from the UAV to the SBS The dedicated sensing signal The duration in the first phase The duration in the phase of collecting the sensed target data The duration in the phase of converting the sensed data The duration in the second phase The maximum total latency of the system The maximum latency of the first phase The maximum latency of the second phase The UAV’s sensing receive beamforming The average velocity of the rotor The transmit beamforming for the based station The weight of the UAV The transmitted sensing signal of the UAV The transmitted signal of USV n The transmitted signal of the UAV in second phase The received signal at the UAV The received signal at the SBS The GPU power-efficiency of the UAV The duty factor The pulse duration The total interference of the UAV for sensing the target The total interference of the UAV for receiving USV n’s data
and one UAV equipped with Nt transmit antennas and Nr receive antennas. The UAV employs ISAC technology to simultaneously receive data from the USVs and sense a target node. Specifically, it performs radar sensing towards a given target located at angle θ0 , with a required estimation accuracy denoted by Qreq . The UAV transmits both the sensed information and the data collected from the USV to the SBS via
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transmission method also employed in [2], and perform target sensing toward angle θ0 , from which raw sensing data are processed into perception information. 1) Signal Model: The transmitted signal of USV n is written as xn =
√ pn sn , ∀n ∈ N ,
(1)
in which pn stands for the transmission power used to send the information symbol sn , and this symbol satisfies E{|sn |2 } = 1. The transmitted sensing signal of the UAV within the first operational stage is formulated as x0 = s0 ,
(2)
Nt ×1
Fig. 1. The MIoT network system model with ISAC. Part A presents the UAV-assisted data collection and target sensing phase, which includes 1 the duration of UAV’s data collection from USVs, 2 the duration of Sensed Target Data Collection, and 3 the duration of Collected Sensed Data-to-Transmitted Data Conversion. 1 is carried out simultaneously with 2 and 3 . Part B presents the UAV-assisted data forwarding phase. wireless communication. We adopt a 3-D Cartesian coordinate framework to describe the spatial positions of all USVs and the UAV, and fix the vertical coordinate of the sea surface to zero. Let qn = (xn , yn , zn ) represent the coordinate of USV n with xn yn and zn denoting its x-coordinate, y-coordinate, and z-coordinate respectively. The UAV’s coordinate is recorded as U = (X, Y, H) where X and Y stand for horizontal positions and H refers to its steady hovering height. Similarly, we mark the coordinate of the SBS as S = (XS , YS , ZS ) where XS , YS and ZS correspond to its three-dimensional spatial dimensions respectively. Table II lists the key notations used in this paper. For clarity, the operational process illustrated in Fig. 1 is split into two independent stages. • During the duration in the UAV-assisted data collection and target sensing phase T1 , the USVs transmit their data to the UAV, while the UAV simultaneously emits a sensing signal toward the target. We adopt the NOMA technique, so the UAV can capture the reflected sensing echo and the transmission signals sent from every USV at the same time. The raw sensing data are subsequently processed into meaningful perception information. • During the duration in the UAV-assisted data forwarding phase T2 , the UAV transmits both the data collected from the USV and the processed sensing information of the target node to the SBS. A. UAV-assisted Data Collection and Target Sensing Phase In this phase, the UAV leverages NOMA to simultaneously collect data {Dn }n∈N from all USVs, where USVs adopt NOMA transmission to send data to the UAV concurrently, a
where s0 ∈ C denotes the dedicated sensing signal with covariance S0 = E{s0 (s0 )H }. We establish the sensing channel model linking the UAV and the sensing target. We suppose that the UAV’s transmitting and receiving uniform linear antenna arrays adopt half-wavelength element separation. Accordingly, the transmit steering vector and receive steering vector corresponding to angle θ take the following forms 1 at (θ) = √ [1, ejπ sin θ , . . . , ejπ(Nt −1) sin θ ]T , Nt
(3)
1 ar (θ) = √ [1, ejπ sin θ , . . . , ejπ(Nr −1) sin θ ]T . Nr
(4)
We consider that there exist M clutters under the angles of {θm }m∈{1,2,...,M} , and the UAV has the prior knowledge of the target and clutters. Given the air-to-sea transmission environment, the transmission link connecting each USV and the UAV, denoted as gnt2 ∈ CNr ×1 , follows Line-of-Sight (LoS) transmission characteristics. Then, the channel coefficient linking USV n to the UAV takes the form gnt2 = h0 √ · ar (θn ), where h0 stands for the LoS channel 2 2 kU−qn k +H
gain at the reference distance d0 = 1 m. Therefore, the received signal at the UAV can take the following form y0 =
N X
√ gnt2 pn sn + β0 ar (θ0 )(at (θ0 ))H x0 | {z } n=1 {z } sensing echo from the desired target |
signal from all USVs
+
M X m=1
|
βm ar (θm )(at (θm ))H x0 + {z
clutter interference
}
H x0 + n0 } | SI {z
,
self-interference and noise
(5) where the complex amplitudes {βm }m=0,1,2,...,M are mainly decided by multiple factors including path loss and radar cross-section. HSI ∈ CNr ×Nt stands for the residual selfinterference channel coefficient existing on the UAV receiver. n0 denotes the noise with variance σ02 . For clarity of notation, we further define As = β0 ar (θ0 )(at (θ0 ))H and Ac = P M H m=1 βm ar (θm )(at (θm )) + HSI . 2) Communication Model: Within this operational stage, the UAV adopts the NOMA technique to recover incoming signals sent by all USVs. We denote the decoding order of the USV data streams by a permutation Θ. For instance, Θ(m) < Θ(n) indicates that the data from USV m is decoded and subtracted from the received waveform prior to processing the
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information of USV n. Due to practical hardware constraints, the perfect SIC is hard to achieve, leading to the residual interference from the direct link. According to [35], [36], the imperfect SIC effect can be effectively characterized by a linear function model. Thus, the achievable transmission rate RnUSV for the uplink transmission from USV n to the UAV takes the subsequent form: RnUSV = B UAV log(1 +
t2 t2 H pn cH n gn (gn ) cn ), ∀n ∈ N , cH n Γn cn
(6)
where B UAV denotes the transmission channel bandwidth in this phase, cn ∈ CNr ×1 represents the receiving beamforming vector deployed on the UAV for communication, and Γn refers to the aggregate interference encountered while decoding the data stream of USV n, and this interference term takes the following form Γn =
X
t2 t2 H H pm gm (gm ) + As S0 AH s + Ac S0 Ac
Θ(m)>Θ(n)
+ σ02 IN + ζ
X
t2 t2 H pm gm (gm ) , ∀n ∈ N .
(7)
Θ(m)<Θ(n)
where ζ represents the SIC factor that quantifies the residual decoded interference in NOMA. Specifically, ζ = 1 means that no SIC is conducted, while ζ = 0 denotes the perfect SIC scenario. The specific value of ζ depends on hardware implementation. Considering the research focus of this paper, ζ is set to 0 in the following derivations. According to [28], the optimal receiving vector c∗n takes the following form t2 t2 H pn cH n gn (gn ) cn c∗n = arg max H cn Γn cn
= Γ−1 n gn , ∀n ∈ N .
(8)
We plug the solved optimal vector c∗n into Eq. (6), so we can restate the attainable uplink data rate in a new expression t2 RnUSV = B UAV log(1 + pn (gnt2 )H Γ−1 n gn ), ∀n ∈ N .
(9)
With the duration T1 , the achievable transmission rate should satisfy (10) T1 RnUSV = Dn , ∀n ∈ N . The total energy consumption generated by USV n takes the form (11) EnUSV = pn T1 , ∀n ∈ N . 3) Sensing Model: To handle the sensing module, we divide the duration T1 into two parts show in Fig. 1, the duration time T11 for collecting the sensed target data and the duration time T12 for converting the collected sensed data into transmitted data, so the duration T1 can be expressed as T1 = T11 + T12 .
(12) 1
We utilize the sensing estimation rate [28] to measure the radar sensing performance, which characterizes the elimination 1 The sensing estimation rate measures the reduction rate of target parameter uncertainty per second. It is suitable for the real-time ISAC systems with latency constraints. Unlike the CRB, which gives a lower bound of estimation error variance for static parameter estimation, the sensing estimation rate emphasizes dynamic sensing efficiency, which aligns better with our marine IoT scenario.
degree of target parameter uncertainties within each unit time. Based on Eq. (5), after decoding the transmission signal from all USVs, the sensing estimation rate Q of the UAV for sensing the target in this phase can be expressed as 2τ B UAV uH As S0 (As )H u δ (13) log(1 + ) 2τ uH Λu in which δ stands for the duty cycle, while τ refers to single pulse time length. The vector u denotes the UAV’s sensing receive beamforming. In Eq. (13), Λ denotes the total interference of the UAV for sensing the target in this phase, and this interference term is formulated as Q=
2 Λ = Ac S0 AH c + σ0 IN .
(14)
Deriving from the research outcome of reference [28], the optimal receiving beamforming vector u∗ takes the following form uH As S0 (As )H u (15) = Λ−1 ar (θ0 ). uH Λu We substitute the solved optimal vector u∗ into Eq. (13), which allows us to restate the target sensing estimation rate as u∗ = arg max
δ −1 (16) log(1 + 2τ B UAV (s0 )H AH As s0 ). s Λ 2τ With the duration T11 , the energy consumption produced by the UAV platform is formulated in the below expression: Q=
UAV E11 = Tr(S0 )T11 .
(17)
4) Computational model: After receiving the sensing information, the UAV processes and converts it into transmissionready data Dtarget through techniques such as feature extraction, compression, and encoding. It can be expressed as Dtarget = αQT11 ,
(18)
where α ∈ (0, 1] denotes the compression efficiency coefficient [37], a dimensionless parameter that quantifies the relationship between the volume of sensing data received by the UAV and the sensing data transmitted by the UAV. The duration T12 can be expressed as T12 =
QT11 C UAV , f UAV
(19)
in which f UAV corresponds to the CPU operating frequency of the UAV, and C UAV denotes the UAV’s required CPU cycles for each bit. With the duration T12 , the total energy consumption generated by the UAV platform is expressed in the below form: UAV E12 = T12 ω UAV (f UAV )3 = ω UAV QT11 C UAV (f UAV )2 , (20)
where ω UAV is the GPU power-efficiency of the UAV. B. UAV-assisted Data forwarding Phase After collecting all data from the USVs and processing the sensing data, the UAV forwards the results to the SBS. 1) Signal Model: The transmitted signal by the UAV within this operational stage takes the form e x0 = wz,
(21)
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in which w ∈ CNt ×1 stands for the transmit beamforming vector dedicated to the SBS, and z satisfying E{|z|2 } = 1 corresponds to the data symbol intended for the SBS. We assume the communication link connecting the UAV and SBS follows line-of-sight transmission, and its channel coefficient h0 takes the form h = √ · at (θ). The signal captured 2 2 kU−Sk +H
by the SBS receiving terminal adopts the following expression e yBS = hH wz + nBS ,
(22)
in which h ∈ CNt ×1 stands for the channel coefficient over the air link between the UAV platform and ground SBS, while 2 nBS refers to the additive Gaussian noise with variance σBS . 2) Communication Model: In this phase, the transmission rate RUAV achieved by the SBS can be expressed as |hH w|2 ), RUAV = B BS log(1 + 2 σBS
(23)
where B BS denotes the transmission channel bandwidth between the UAV and the SBS. With the duration T2 , the transmission data should satisfy N X n=1
Dn + Dtarget ≤ RUAV T2 .
(24)
During this phase, the total energy consumption generated by the UAV platform is formulated in the below expression: E
UAV
H
= Tr(ww )T2 .
(25)
C. Total Energy Consumption Before calculating the overall energy consumption of the system, we need to take the hovering energy consumption of the UAV within the two time segments into consideration, and this energy term takes the following form E UAV,hov = phov (T1 + T2 ),
(26)
hov
in which p stands for the steady power consumed to maintain the UAV’s hovering state, and this parameter is mainly determined by multiple factors including the aircraft weight, air density, rotor efficiency and lift magnitude. According to [38], [39], it can be expressed as 3/2
wUAV , phov = √ 2ρACT
(27)
where wUAV denotes the weight of the UAV, ρ refers to the ambient air density, and A represents the rotor disc area, which is typically expressed as the square of the rotor radius r multiplied by π, i.e., A = πr2 , and CT denotes the thrust coefficient. Notably, most existing works adopt the ideal momentum theory model with CT = 1 to simplify UAV hovering energy consumption by neglecting the thrust coefficient CT . However, practical rotors exhibit aerodynamic losses such as blade element drag and uneven airflow. Following [40], we introduce CT to refine the ideal model, thereby enhancing the accuracy of hovering power characterization. It can generally be expressed as L CT = 1 , (28) 2 2 ρAv
where L denotes the lift force provided by the rotor, which is typically equal to the weight of the UAV wUAV in a hovering state, and v denotes the average velocity of the rotor. According to Eqs. (26), (27), and (28), the hovering energy consumption of the UAV can be rewritten as 1 UAV (29) vw (T1 + T2 ). 2 The overall energy consumption generated by the whole system is E UAV,hov =
E tot =
N X
UAV UAV EnUSV + E11 + E12 + E UAV + E UAV,hov . (30)
n=1
D. Problem Formulation This work targets the minimization of the system energy consumption while guaranteeing both the latency requirement and the sensing rate requirement. To reach this optimization objective, we jointly optimize the transmit beamforming of the UAV w, the dedicated sensing signal s0 , the transmit powers of the USVs {pn }n∈N , the computation power of the UAV f UAV , and the durations T11 and T2 . The formulated optimization problem is presented below: P1: min E tot s.t. constraints (10), (24), T11 + T12 + T2 ≤ T tot,max ,
(31a)
Q ≥ Qreq , 0 ≤ T1 ≤ T1max ,
0 ≤ T2 ≤ T2max , H
Tr(ww ) ≤ p
Tr(S0 ) ≤ p
(31d)
UAV,max
UAV,max
(31b) (31c)
,
(31e)
,
(31f)
pn ≤ pUSV,max , n
(31g)
f UAV,min ≤ f UAV ≤ f UAV,max ,
vars. s0 , w, {pn }n∈N , T11 , f
UAV
(31h) , T2 .
Constraint (31a) ensures that the total latency doesn’t exceed the maximum allowable latency T tot,max . Constraint (31b) guarantees the required sensing estimation rate Qreq . Constraints (31c) and (31d) mean that both the duration in Phase-I T1 and the duration in Phase-II T2 cannot exceed a specified upper-bound. Constraints (31e) and (31f) cap the total transmission power radiated by the UAV within its hardware power limit pUAV,max . Constraint (31g) provides an upper bound for USV’s transmit power. Constraint (31h) implies that the UAV’s processing-rate cannot exceed its computing capacity f UAV,max . IV. A LGORITHM D ESIGN In this section, we tackle the optimization problem P1 via the decomposition framework. In detail, we first derive closed-form solutions for all USV transmit power variables and convert the problem P1 into an equivalent problem P2. Next, we split P2 into two nested subproblems through vertical decomposition: the lower-layer subproblem P2-L,
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We rewrite E tot as E
tot
=
Dn
N X
C −1 (T11 + QT11UAV f (2 (gnt2 )H Ω−1 gnt2
UAV
)B UAV
n=1
P
Θ(m)>Θ(n) Dm QT11 C UAV )B UAV f UAV
− 1)
QT11 C UAV ) f UAV + Tr(S0 )T11 + ω UAV QT11 C UAV (f UAV )2 + Tr(wwH )T2 1 QT11 C UAV + vwUAV (T11 + + T2 ), 2 f UAV (33) H 2 where Ω = As S0 AH + A S A + σ I . c 0 N s c 0 Then, we can rewrite the original optimization formulation P1 into its equivalent counterpart labeled P2, which is presented below: 2
(T11 +
(T11 +
P2: min E tot s.t. constraints (24), (31a), (31b), (31c), (31d), (31e), (31f), (31h), Fig. 2. Decomposition Structure for Solving Problem P1.
Dn
C −1 (T11 + QT11UAV f (gnt2 )H Ω−1 gnt2 (2
UAV
P
which optimizes the UAV’s transmit beamforming, the dedicated sensing signal, and computation power, and a upper level problem P2-U, which optimizes the durations in the two phases. To solve P2-L, we apply the block coordinate descent (BCD) method [28], [41], [42] and further decompose it horizontally into three separate subproblems: P2-L-f for tuning the UAV’s computation power, P2-L-w for designing the UAV transmit beamforming and P2-L-s for optimizing the dedicated sensing signal. We put forward an algorithm built on polyblock approximation to handle the optimization subproblem P2-L-f . For problem P2-L-w, we leverage the inherent rank-1 characteristic of the transmit beamforming and construct an alternating optimization algorithm for its solution. For P2-L-s and P2-U, we apply the SCA method and solve them iteratively using the CVX solver. This multilayer solution framework enables us to efficiently resolve the original optimization P1 through iterative iterations, as illustrated in Fig. 2.
2
Θ(m)>Θ(n) Dm QT11 C UAV (T11 + )B UAV f UAV
vars. s0 , w, T11 , f
UAV
≤ pUSV,max , n
)B UAV
− 1) (34a)
, T2 .
B. Decomposition of Problem P2 The reformulated optimization formulation P2 still falls into the category of strictly non-convex optimization problem. To address this difficulty, we put forward a layered decomposition strategy as the core solution. 1) Lower level problem under the given T11 and T2 : We firstly consider that the durations T11 and T2 are given, and aim at optimizing the transmit beamforming of the UAV w, the dedicated sensing signal s0 , and the computation power of the UAV f UAV . This leads to problem P2 turns into problem P2-L as follows: P2-L: min E tot s.t. constraints (24), (31a), (31b), (31c), (31e), (31f), (31h), (34a) vars. s0 , w, f UAV .
A. Problem Transformation We can observe that the optimization formulation P1 falls into the category of strictly non-convex optimization problems. This inherent property brings obstacles to directly solving the original formulation P1. To address this difficulty, we first restructure the mathematical expression of P1 to derive its equivalent optimization form. By putting Eq. (9) into constraint (10), we can derive H 2 −1 t2 −1 pn = (gnt2 )H (As S0 AH gn s + Ac S0 Ac + σ0 IN ) (2
Dn QT11 C UAV )B UAV (T11 + f UAV
P
− 1)2
Θ(m)>Θ(n) Dm QT11 C UAV )B UAV (T11 + f UAV
. (32)
2) Upper level problem to optimize T11 and T2 : After solving problem P2-L and obtaining the value of E tot with the given T11 and T2 , we then continue to optimize T11 and T2 , which results in the following optimization problem. P2-U: min E tot s.t. constraints (24), (31a), (31c), (31d), (34a), vars. T11 , T2 . C. Decomposition of Problem P2-L In problem P2-L, the UAV’s CPU frequency f UAV , the UAV’s transmit beamforming w, and the dedicated sensing signals s0 show a strong coupling effect within both the objective term and all constraint conditions. To acquire an efficient
8
solution for subproblem P2-L, we propose the following decomposition of problem P2-L based on the BCD method. 1) Sub-problem to optimize f UAV under the given w and s0 :
With the implicit monotonic property of subproblem L-f1 revealed in Proposition 1, we can establish problem L-f1 as a canonical monotonic optimization form by introducing an auxiliary variable e and a new function H(x, e), which are shown as follows:
P2-L-f : min E tot s.t. constraints (31a), (31c), (31h), (34a), var. f UAV . 2) Sub-problem to optimize w under the given f UAV and s0 : P2-L-w: min E tot s.t. constraints (24), (31e), 3) Sub-problem to optimize s0 under the given f UAV and w: P2-L-s: min E tot s.t. constraints (24), (31a), (31b), (31c), (31f), (34a), var. s0 . According to the criterion of the BCD method, the three sub-problems (i.e., problem P2-L-f , problem P2-L-w and problem P2-L-s) are solved iteratively to acquire the solution of the lower-layer formulation P2-L. This splitting scheme possesses a primary advantage in that all decomposed subproblems can be handled separately to attain their respective optimal points. D. Proposed Algorithm for Solving Problem P2-L-f C We introduce the auxiliary variable x = T11 + QTf11UAV and problem P2-L-f can then be transformed into
L-f1: min
(gnt2 )H Ω−1 gnt2
n=1
2
P Θ(m)>Θ(n) Dm xB UAV
−1
UAV
,
Dn
(2 xBUAV − 1)
x
+ Tr(S0 )T11 + ω UAV QT11 C UAV (
QT11 C UAV 2 ) x − T11
1 + Tr(wwH )T2 + vwUAV (x + T2 ) 2 s.t. x ≤ T tot,max − T2 , (35a) UAV QT11 C x ≥ UAV,max + T11 , (35b) f P Θ(m)>Θ(n) Dm −1 Dn xB UAV (2 xBUAV − 1)2 (gnt2 )H Ω−1 gnt2 ≤ pUSV,max , n max 0 ≤ x ≤ T1 ,
UAV e = E max − (E UAV,hov + E11 + E UAV ), UAV H(x, e) = e − (E12 +
var. w.
N X
Proposition 1. The objective function of problem L-f1 is the difference between two monotonic functions with respect to x.
N X
EnUSV ).
(36) (37)
n=1
Here, E max can be acquired when variable x is assigned its upper bound xmax , and the subproblem L-f1 can be rewritten into its equivalent formulation presented below: L-f2: max H(x, e) s.t. constraints (35c) QT11 C UAV max{0, UAV,max + T11 } ≤ x f ≤ min{T1max, T tot,max − T2 }, 0≤e≤E
max
− (E
UAV,hov
(38a)
UAV + E11 + E UAV ),
(38b)
var. x, e. We can observe that the mapping H(x, e) rises monotonically as variables x and e grow separately. When H(x, e) attains its maximum value, the objective term of subproblem L-f1 reaches its minimal value, and the corresponding value of x delivers the optimal decision variable. Based on the constraints (35c), (38a) and (38b), we define the normal cone set X and conormal cone set Y as presented below: X = {(x, e)|constraints (38a) and (38b)}.
(39)
Y = {(x, e)|constraint (35c)}.
(40)
Exploiting the feature of problem L-f2 in Proposition 1, we put forward Algorithm 1 built upon polyblock approximation from references [28], [43]. This algorithm aims to search for the optimal pair (x∗ , e∗ ) that minimizes the value of H(x, e).
E. Proposed Algorithm for Solving Problem P2-L-w
(35c)
(35d)
var. x.
We analyze the monotonic characteristic of the objective function belonging to subproblem L-f1 with respect to x, as shown in Proposition 1. Full derivations to verify Proposition 1 are provided within Appendix A.
To solve problem P2-L-w, we first define W , wwH . Here, W stands for a positive semi-definite matrix satisfying W 0 with unit rank, and we also introduce H , hhH . With this definition, constraint (24) can be rewritten as Tr(HW) ≥ (e
PN target n=1 Dn +D B BS T2
2 − 1)σBS .
(41)
9
Algorithm 1 To Solve Problem L-f2 and Obtain Optimal (x∗ , e∗ ) = {(xmax , emax )}. Set the current best value CBV0 = −∞ and the current best solution CBS0 as an empty set. Set the iteration number i = 0. Set ν as a very small positive number. 2: repeat 3: Update i = i + 1 4: Find the vertex zi = arg max{H(x, e)|(x, e) ∈ Vi−1 } from Vi−1 . 5: Use bisection search to find the projection of zi on the boundary of X , which is denoted as πX (zi ). 6: if |πX (zi ) − zi | ≤ ν then 7: Update CBSi = zi and CBVi = H(zi ). 8: else 9: if πX (zi ) ∈ X ∩ Y and H(πX (zi )) ≥ CBVi−1 then 10: Update CBSi = πX (zi ) and CBVi = H(πX (zi )). 11: else 12: Update CBSi = CBSi−1 and CBVi = CBVi−1 . 13: end if 14: Set ztem = πX (zi ) and update Vi = {Vi−1 \zi } ∪ {(zi (1), ztem (2)), (z n (1), ztem (2))}. 15: Remove from Vi the improper vertices and the vertices {v ∈ Vi |v ∈ / Y}. 16: end if 17: until |H(zi ) − CBVi | < ν 18: Output: the optimal solution (x∗ , e∗ ) = CBSj for problem L-f2. Notice that x∗ is also the solution for problem L-f1. 1: Initialize the vertex set V0
matrix W. From the conclusion of Proposition 2, we can efficiently handle the subproblem L-w2 via mature convex optimization solvers such as CVX. We conduct iterative computation for e Then, we L-w2 and mark the converged output matrix as W. e construct a Rank-1 matrix solution based on W to acquire the optimal solution of subproblem P2-L-w, which are presented below: e e − 12 Wh. (44) w∗ = (hH Wh) The details for solving problem P2-L-w are presented in Algorithm 2.
Algorithm 2 To Solve Problem P2-L-w and obtain w∗ 0
1: Initialize W . Set the iteration number t=0. Set ∆ as a
very small positive number. 2: repeat 3: Update t = t+1.
Solve problem L-w2 with Wt−1 by CVX and obtain Wt 5: until Convergence. 6: Calculate w∗ according to Eq.(44). 7: Output: the solution w∗ for problem P2-L-w. 4:
F. Proposed Algorithm for Solving Problem P2-L-s H To solve problem P2-L-s, we define X , AH s u, Y , Ac u and Z , σ02 uH u. Then, constraints (31a), (31b) and (31c) can be rewritten as
Then, problem P2-L-w can be equivalently reformulated as follows: L-w1: min E
tot
XH S0 X ≤ C1 (YH S0 Y + Z),
(45)
XH S0 X ≥ C2 (YH S0 Y + Z),
(46)
XH S0 X ≤ C3 (YH S0 Y + Z),
(47)
where
s.t. constraint (41), UAV,max
Tr(W) ≤ p Rank(W) = 1,
,
C1 =
(42a) (42b)
var. W.
UAV τ 1 ) (T tot,max −T11 −T2 )( 2f UAV T11 C δ − 1)), (2 2τ B UAV 2τ Qreq 1 C2 = (2 δ − 1), UAV 2τ B max
Then, we employ the semidefinite relaxation (SDR) technique and temporarily ignore the Rank-1 constraint shown in formula (42b). This treatment enables us to resolve the subproblem L-w1 via iterative computation over the subsequent optimization formulation. L-w2: min E tot Tr(W) ≤ pUAV,max , var. W.
(43a)
Next, we analyze the convex property of the relaxed problem L-w2 with respect to matrix W and summarize the conclusion as Proposition 2. Full derivations to verify Proposition 2 are provided within Appendix B. Proposition 2. The subproblem L-w2 belongs to the class of strictly convex optimization formulations with respect to
(49)
UAV
2(T11 −T11 )f τ 1 T11 C UAV δ (50) (2 − 1), 2τ B UAV The objective function together with constraints (24) and (34a) contains non-convex terms, so we apply SCA to solve them and obtain a more tractable form. We define H 0X , ψ(S0 ) = θ(S0 ) = (gnt2 )H Ω−1 gnt2 , γ(S0 ) = 2τ B UAV YHX S0SY+Z
C3 =
log(1 + γ(S0 )), η1 (S0 ) =
s.t. constraint (41),
(48)
P
Θ(m)>Θ(n) Dm T11 C UAV δ ψ(S0 ) f UAV 2τ
T11 +
PN
T11 +
n=1 Dn T11 C UAV δ ψ(S0 ) f UAV 2τ
, and η2 (S0 ) =
. With these auxiliary mappings defined, we
can restate the objective function and constraints (24), (34a) in the following equivalent form: N X
Dn + αT11 ψ(S0 ) ≤ RUAV T2 ,
(51)
1 (2η1 (S0 ) − 2η2 (S0 ) ) ≤ pUSV,max , n θ(S0 )
(52)
f4 (S0 ) =
n=1
f1 (S0 ) =
10
E tot =
N X
f1 (S0 )(T11 +
n=1
T11 C UAV δ log(1 + γ(S0 ))) + Tr(S0 )T11 f UAV 2τ
+ ω UAV T11 C UAV (f UAV )2 +
δ log(1 + γ(S0 )) + Tr(wwH )T2 2τ
1 UAV T11 C UAV δ vw (T11 + T2 + log(1 + γ(S0 ))). 2 f UAV 2τ
(53)
By analyzing the convex approximation property of the transformed non-convex expressions, we draw the conclusion presented in Proposition 3. Full derivations for verifying Proposition 3 are placed in Appendix C. Proposition 3. We can substitute the original objective function together with constraints (24) and (34a) by the convex objective term and convex constraints listed below f4 (S(k) ) + ▽f4 (S(k) )(S0 − S(k) ) ≤ RUAV T2 ,
(54)
f1 (S(k) ) + ▽f1 (S(k) )(S0 − S(k) ) ≤ pUSV,max , n
(55)
E tot ≈ E tot (S(k) ) + ▽E tot (S(k) )(S0 − S(k) ),
(56)
Therefore, problem P2-L-s can be restated into a standard convex optimization formulation presented below, and we can also use CVX to solve this problem. L-s1: min E tot s.t. constraints (31f), (45), (46), (47), (55), var. S0 . G. Proposed Algorithm for Solving Problem P2-U In problem P2-U, by performing the first-order Taylor P expansion of T11 in f (T11 ) P
=
2
T11 (1+
N n=1 Dn QC UAV )B UAV f UAV
Θ(m)>Θ(n) Dm QC UAV T11 (1+ )B UAV f UAV
2 f (T11 ) (k)
f (T11 ) ln 2
≈ +
−
using the SCA method, we obtain (k) (k) (k) ≈ f (T11 ) + ▽f (T11 )(T11P − T11 )
(−
ln 2
PN
UAV
n=1 Dn (k)
(1+ QCUAV )B UAV (T11 )2
2
(1+
N n=1 Dn (k) QC UAV )B UAV T11 f UAV
+
fP
P
Θ(m)>Θ(n) Dm UAV (k)
(1+ QCUAV )B UAV (T11 )2
2
Θ(m)>Θ(n) Dm (k) QC UAV (1+ )B UAV T 11 f UAV
)(T11
f
−
(k)
T11 ).
Problem P2-U can be transformed into convex problem U-T1, which can be efficiently solved using the CVX optimization toolbox. U-T1: min
N X
(gnt2 )H Ω−1 gnt2
−1
f (T11 )(T11 +
n=1
QT11 C UAV ) f UAV
+ Tr(S0 )T11 + ω UAV QT11 C UAV (f UAV )2 + Tr(wwH )T2 QT11 C UAV 1 UAV vw (T11 + + T2 ) 2 f UAV s.t. constraints (24), (31a), (31c), (31d), −1 USV,max (gnt2 )H Ω−1 gnt2 f (T11 ) ≤ pn , vars. T11 , T2 . +
(57a)
Finally, to solve the original problem P1, we execute Algorithm 3 and problem P2-U alternately until their respective solutions keep unchanged. The details for solving problem P2-L are presented in Algorithm 3. We next analyze the computational complexity of the proposed algorithm. Algorithm 1 is invoked in Step 2 of
Algorithm 3 BCD-based Algorithm to Find the optimal Solution of Problem P2-L 1: repeat 2: Using Algorithm 1 to slove problem P2-L-f and obtain the value of f UAV . 3: Using Algorithm 2 to slove problem P2-L-w and obtain the value of w. 4: until the solutions of problems P2-L-f , P2-L-w and P2-L-s keep unchanged.
Algorithm 3 to solve problem P2-L-f and its complexity is dominated by vertex set update steps (Lines 4, 14, and 15) and bisection search (Line 5). Each vertex set update step has a complexity of O(|V|) where |V| denotes the size of the vertex set in Algorithm 1. The bisection search has a complexity of O log 1ǫ with ǫ as the search precision [44]. Algorithm 2 is invoked in Step 3 of Algorithm 3 to solve problem P2-L-w and its complexity mainly arises from interior-point methodbased solution for the beamforming subproblem, yielding a complexity of O(Nt3 ) where Nt is the number of UAV transmit antennas. The SCA step in Step 4 of Algorithm 3 targets problem P2-L-s and has a complexity of O(Nt3 ) per iteration. Problem P2-U focuses on phase duration optimization and involves two scalar variables which are durations of the two phases, so its complexity is O(1) as a constant term per iteration. Let K denote the number of iterations for Algorithm 3 to converge. The overallcomplexity of Algorithm 3 is thus where the per-iteration cost is O K · |V| + log 1ǫ + Nt3 dominated by the O(Nt3 ) term from beamforming and sensing signal optimization. The stopping criterion for Algorithm 3 is defined to terminate when the relative change in the total energy consumption E tot between consecutive iterations is less than 10−3 . The constant complexity of problem P2-U is combined into the asymptotic analysis. Accordingly, the overall computational complexity of the proposed algorithm is O K · |V| + log 1ǫ + Nt3 . V. S IMULATION R ESULTS This section presents numerical simulations to evaluate the performance of our proposed algorithm and overall system framework. Referring to the IoT network parameters in the literature [35], we set the UAV platform to equip Nt = 10 transmitting antennas and Nr = 15 receiving antennas, the target sensing node is placed at the azimuth angle θ0 = 0◦ , while all clutter scatterers correspond to angular positions {−50◦, −30◦ , 30◦ , 50◦ }. The horizontal separation between the BS and UAV is fixed at 500 m, and all USVs are randomly distributed within a 200 m radius centered on the UAV. We model the wireless links between USVs and the UAV, as well as the connections linking the UAV and SBS, as Rayleigh fading channels with standard path loss attenuation, and we configure core system parameters as B UAV = B BS = 10 MHz for communication bandwidth, δ = 0.01 for the duty cycle factor, τ = 2 × 10−6 s for pulse time length, C UAV = 1 × 103 cycles/bit for the UAV’s per-bit CPU computation overhead, ω UAV = 1 × 10−28 for the UAV GPU power efficiency 2 coefficient, σBS = σ02 as -174 dBm/Hz for background noise
11
power, 1 W as the peak transmit power limit, data payload sizes ranging from 1 Mbits to 2 Mbits, a maximum Phase-I time budget T1max s and a maximum Phase-II time budget T2max s. The self-interference channel matrix at the UAV p dm,n follows the expression [HSI ]m,n = ℵm,n e−j2π λ , where ℵm,n = −110 dB stands for the self-interference channel gain, and dˆm,n denotes the spatial gap separating the m-th receive antenna and the n-th transmit antenna.
LINGO. Numerical outcomes reveal that the average runtime of our proposed algorithm is over 7 times shorter than that of GA and more than 11 times lower compared with LINGO.
A. Convergence and Optimality Performance
The total energy consumption(J)
This subsection carries out verification on the convergence performance and optimality of our proposed optimization algorithm. We consider four scenarios, and the number of USVs is configured as 3, 5, 7 and 9. Fig. 3 illustrates the convergence performance of our proposed optimization algorithm under varying quantities of USVs. We can observe that the algorithm converges within 200 iterations when the USV count equals 5 or 7, while approximately 250 iterations are required to reach convergence for cases with 3 or 9 USVs Moreover, total system energy expenditure rises as the number of USVs increases. This is because, under the same latency constraint, a larger number of USVs results in a higher volume of data to be transmitted, thereby requiring more energy to satisfy the offloading demand.
Fig. 4. The accuracy of the proposed algorithms in comparison with GA and LINGO.
4 The number of USVs =3 The number of USVs =5 The number of USVs =7 The number of USVs =9
3 2 1
Fig. 5. The average running time per time slot of the proposed, GA algorithms and LINGO. 0
0
200
400
600
800
The number of iterations
Fig. 3. The convergence performance for different numbers of USVs. In Fig. 4, we compare the total system energy consumption of the proposed algorithm, the genetic algorithm (GA)2 , and linear interactive and general optimizer (LINGO)3 under different numbers of USVs. The proposed algorithm outperforms the GA, with an average deviation in total energy consumption of no more than 8%, and stays within 8.72% of the result obtained by LINGO. In Fig. 5, we carry out comparisons of average runtime among our proposed algorithm, GA and 2 GA is the comparison benchmark since it is a well-established heuristic for complex optimization, capable of exploring solution spaces to yield expected feasible solutions, which enables valid performance verification of the proposed algorithm. 3 LINGO is a professional optimization tool that can provide high-precision solutions for nonlinear programming problems, so we adopt it as a benchmark to verify the optimality of the proposed algorithm.
B. Performance of the Proposed Scheme This subsection assesses the overall performance of our proposed scheme via cross-checks against multiple benchmark schemes. Fig. 6 delivers a comparative analysis of overall system energy consumption between our proposed NOMA-based transmission scheme and the Orthogonal Frequency Division Multiple Access (OFDMA) scheme. The results show that our NOMA-based scheme can reduce the system energy consumption by 19.71% compared to the OFDMA scheme. Since NOMA superimposes multiple USV signals within the same time-frequency resource block and employs SIC technology, it effectively improves spectral efficiency and thus reduces system energy consumption. It implies that battery-limited USV fleets can enjoy prolonged endurance and an extended operating range. Fig. 7 further illustrates that the overall energy consumption decreases as the maximum allowable duration in Phase I
The total energy consumption(J)
12
2
No. of USVs=3 No. of USVs=5 No. of USVs=7 No. of USVs=9
1.5 1 0.5 0
1
1.5
2
2.5
3
The required sensing estimation rate (Kbits/s) Fig. 8. The total energy consumption under different required sensing estimation rates.
increases. Moreover, as the number of USVs increases from 3 to 9, the proposed NOMA-based scheme consistently achieves lower energy consumption than that of the OFDMA-based scheme. Specifically, the NOMA can reduce the total energy consumption by 13.5% on average, in comparison with the OFDMA. This is because that a longer allowable duration in Phase I lets each USV lower its transmit power, so the total energy consumption decreases.
the total energy consumption increases with as the distance between USV and UAV increases. This is because that farther distance leads to higher path loss and degraded channel quality. It implies that the same data rate requirement should be satisfied at the cost of each USV to transmit with higher power or longer duration for the USV.
The total energy consumption(J)
1.8
NOMA scheme with no. of USVs=3 NOMA scheme with no. of USVs=5 NOMA scheme with no. of USVs=7 NOMA scheme with no. of USVs=9 OFDMA scheme with no. of USVs=3 OFDMA scheme with no. of USVs=5 OFDMA scheme with no. of USVs=7 OFDMA scheme with no. of USVs=9
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2
The total energy consumption(J)
Fig. 6. The total energy consumption obtained by NOMA, and OFDMA at the different USV densities.
2 1.5
No. of USVs=3 No. of USVs=5 No. of USVs=7 No. of USVs=9
1 0.5 0
100
200
300
400
500
The distance between the USV and the UAV(m) 1
1.5
2
2.5
3
3.5
4
The maximum Phase-I duration (s)
Fig. 7. The total energy consumption under different maximum Phase-I duration. Fig. 8 plots the total system energy consumption with different required sensing estimation rates, when varying the number of USVs varies from 3 to 9. It can be seen that given the number of USVs, the total system energy consumption always increases with the increase of required sensing estimation rates. This is because that higher sensing rate requirements lead to greater communication demands for each USV, which results in higher transmit power or longer transmission durations. It implies that the energy-aware mission planning should jointly tune sensing quality and USV density to keep the battery budget balanced. Fig. 9 plots the total system energy consumption under different distances between the USVs and the UAV, when varying the number of USVs from 3 to 9. It can be seen that
Fig. 9. The total energy consumption under different distances between the USV and UAV. In Fig. 10, we compare the proposed scheme with four benchmark schemes, the maximum dedicated sensing signal scheme, the maximum UAV’s transmit beamforming scheme, the maximum USV’s transmit power scheme, and the maximum UAV’s CPU frequency scheme, in terms of energy consumption. We can find that, compared to the four baseline schemes, the proposed scheme achieves average reductions in energy consumption by factors of 1.2, 1, 0.7, and 2, respectively. These results demonstrate that we can significantly improve the energy efficiency and reduce the overall system energy expenditure by jointly optimizing the dedicated sensing signal, UAV’s transmit beamforming, UAV’s transmit powers, and UAV’s CPU frequency. It implies that our proposed approach can be applied to scenarios with energy-constrained UAV fleets, and UAVs can extend their mission duration or sense more targets without extra battery power supplies.
13
VI. C ONCLUSION
Fig. 10. The total energy consumption under different optimization schemes. As we know, we can simplify the solution of energy consumption minimization by assuming the perfect interference cancellation and SIC decoding order at the cost of performance degradation. Thus, in Fig. 11, we carry out numerical simulations to test the robustness of our proposed scheme under scenarios with residual NOMA interference and mismatched SIC decoding orders. Fig. 11 records the overall energy consumption achieved by our proposed scheme within the two imperfect channel setups. Numerical observations reveal that when our proposed scheme is applied to these two imperfect scenarios, it exhibits slight performance loss. Specifically, the loss is only 9.66% and 12.52% for the residual interference and the mis-specified SIC order, respectively. Since more optimal energy consumption is needed to mitigate the co-channel interference in the imperfect condition than that in the perfect condition, the performance loss is less when compared to the optimal energy consumption needed in these imperfect scenarios. This result confirms that our proposed scheme can be effectively applied to imperfect scenarios with low computational complexity.
Fig. 11. The total energy consumption of our proposed scheme compared with residual NOMA interference and mis-specified SIC order
This paper investigates the energy consumption minimization problem for an ISAC system in MIoT. In this system, the UAV performs target sensing and data collection from USVs during the first phase, and subsequently forwards both the sensing and the data collected from the USVs to the SBS in the second phase. We formulate a joint optimization problem covering the UAV’s transmit beamforming, dedicated sensing signal design, USVs’ transmit powers, the computation power of the UAV, and the durations in the sensing and communication phases, with the objective of minimizing overall system energy consumption. Although the proposed joint optimization problem is non-convex, we develop a decomposition-based framework with corresponding algorithms to solve it effectively. Numerical outcomes verify the accuracy and efficiency of our proposed optimization algorithm. For future research directions, we will address the limitations of this single-target sensing model and static channel assumption. We will focus on multi-target sensing scenarios to extend the applicability of the proposed ISAC scheme to complex marine environments.
A PPENDIX A P ROOF OF P ROPOSITION 1 The objective function of subproblem L-f1 falls into five namely all USVs’ energy consumption PN components, USV for transmitting data, the UAV’s energy consumpE n=1 n UAV UAV and data , computing data E12 tion for sensing data E11 UAV forwarding E , the UAV’s energy consumption for hovering E UAV,hov . According to Eq. (12) and Eq.(29), it can be observed that E UAV,hov exhibits an upward trend as the variable x rises. UAV UAV E11 and E UAV are constants with respect to x, and E12 is decreasing with respect to x. We then prove that the cumulative PN USV transmission energy n=1 EnUSV decreases as x grows. The first-order derivative of EnUSV with respect to x can be expressed as A+B dEnUSV A+B B B = C −1 2 x (1 − ln 2 ) − 2 x (1 − ln 2 ) dx x x = C −1 (h(A + B) − h(B)) (A.1) P Θ(m)>Θ(n) Dm Dn t2 H −1 t2 where A = B UAV , B = , C = (gn ) Ω gn B UAV a and h(a) = 2 x (1−ln 2 xa ), x > 0. We can obtain the first-order derivative of h(a) with respect to a as (ln 2)2 a dh(a) (A.2) = −a 2 x < 0. da x2 By (A.2), we conclude that h(a) monotonically declines as the variable a rises. Accordingly, we obtain h(A+ B)− h(B) < 0 dE USV in (A.1). With (A.2) and C > 0, we can derive that dxn < 0, PN USV which proves that is decreasing with respect n=1 En to x. Thus, we can reorganize the objective expression of formulation L-f1 into UAV UAV E tot = E UAV,hov + E11 + E UAV − (E12 +
N X n=1
EnUSV ), (A.3)
14
which corresponds to the gap between two functions that both rise monotonically with the increase of x. Therefore, Proposition 1 is proved.
following approximation E tot =
N X
f2 (S0 ) + Tr(S0 )T11 + ω UAV T11 C UAV (f UAV )2
n=1
+ Tr(wwH )T2 +
A PPENDIX B P ROOF OF P ROPOSITION 2
1 We first take the first-order derivatives of θ(S) , γ(S), 2η1 (S) , and 2η2 (S) as follows: 1 ′ 1 −1 t2 t2 H −1 ( ) =− (−AH gn (gn ) Ω As s Ω θ(S) (θ(S))2 (C.1) −1 t2 t2 H −1 − AH Ω g (g ) Ω A ), c c n n
(2
2τ B UAV (YH SY + Z)XXH − (XH SX)YYH , 2 (Y SY + Z) (C.2) ′ H
η1 (S) ′
) =2
η1 (S)
PN
T11 C UAV δ (γ(S)) n=1 Dn f UAV 2τ 1+γ ln 2 , T11 C UAV δ (T11 + f UAV 2τ log(1 + γ(S)))2
−
(C.3) ′ T11 C UAV δ (γ(S)) D Θ(m)>Θ(n) m f UAV 2τ 1+γ η2 (S) ′ η2 (S) . (2 ) =2 ln 2 C UAV δ 2 (T11 + Tf11UAV 2τ log(1 + γ(S))) (C.4) Then, we define constraints (24) and (34a) as f (S 4 0) = PN 1 η1 (S0 ) η2 (S0 ) (2 −2 ), D +αT ψ(S ) and f (S ) = 11 0 1 0 n=1 n θ(S0 ) by performing the first-order Taylor expansion on f4 (S0 ) and f1 (S0 ) , we obtain the following approximation −
P
f4 (S0 ) ≈ f4 (S(k) ) + ▽f4 (S(k) )(S0 − S(k) ),
(C.5)
f1 (S0 ) ≈ f1 (S(k) ) + ▽f1 (S(k) )(S0 − S(k) ),
(C.6)
▽f4 (S(k) )
where
(k)
▽f1 (S
)
=
(k) 1 (2η1 (S ) − 2 θ(S(k) )
(k) 1 αT11 1+γ(S )) , (k) ) (γ(S ′
=
′ (k) ( θ(S1(k) ) ) (2η1 (S ) η2 (S(k) ) ′
).
− 2
η2 (S(k) )
and ) +
Next, we denote f2 (S0 ) = θ(S10 ) (2η1 (S0 ) − 2η2 (S0 ) )(T11 + T11 C UAV δ f UAV 2τ log(1 + γ(S0 ))). We take the first-order derivative of f2 (S0 ) as follows: ′
1 ′ η1 (S0 ) T11 C UAV δ ) (2 − 2η2 (S0 ) )(T11 + f3 (S0 )) θ(S0 ) f UAV 2τ ′ 1 T11 C UAV δ + (2η1 (S0 ) − 2η2 (S0 ) ) (T11 + f3 (S0 )) θ(S0 ) f UAV 2τ ′ T11 C UAV δ 1 (2η1 (S0 ) − 2η2 (S0 ) )(T11 + f3 (S0 )) , + θ(S0 ) f UAV 2τ (C.7) ′
(f2 (S0 )) = (
(γ(S0 )) . By performwhere (f3 (S0 )) = (log(1 + γ(S0 ))) = 1+γ(S 0) ing the first-order Taylor expansion on E tot , we obtain the ′
′
(C.8) PN ′ where ▽E tot (S(k) ) = (f (S )) + 2 0 n=1 ′ δ (f3 (S0 )) + T11 + ω UAV T11 C UAV (f UAV )2 2τ ′ 1 UAV T11 C UAV δ Therefore, Proposition 3 is 2 vw f UAV 2τ (f3 (S0 )) . proved.
R EFERENCES
A PPENDIX C P ROOF OF P ROPOSITION 3
′
T11 C UAV δ 1 UAV vw (T11 + T2 + f3 (S0 )) 2 f UAV 2τ
≈ E tot (S(k) ) + ▽E tot (S(k) )(S0 − S(k) ),
The objective function in the problem L-w2 can be determined is strictly convex with respect to W, since Tr(wwH ) = ||w||22 is a convex function. Constraints (41) and (43a) are all affine. Therefore, Proposition 2 follows.
(γ(S)) =
δ f3 (S0 ) 2τ
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