This letter addresses the quadratic equality constrained least squares (QEC-LS) problem, a class of non-convex optimization problems that arise in various signal processing and communication applications. We revisit the alternating direction method of multipliers (ADMM) approach to QEC-LS problem and investigate its convergence and efficiency. Despite the inherent non-convexity, the proposed ADMM algorithm is proved to converge globally only requiring the quadratic term equal to a positive constant. Numerical results demonstrate that our method achieves global optimality with significantly reduced complexity compared to existing approaches such as semidefinite relaxation and primal-dual methods.

Environmental sensing can be achieved using Wireless Sensor Networks (WSNs) with low-complexity sensors. Integrated sensing and communication (ISAC) allows simultaneous sensing and data transmission through optimized waveform, architecture, and protocol design. Given the multiple sensors in WSNs, minimizing interference and ensuring Doppler-tolerant pulses for detecting dynamic targets is crucial. This paper proposes a Doppler-tolerant waveform design that embeds multiple communication symbols within each pulse of a coherent processing interval (CPI). We formulate a Sum-of-Squares (SoS) optimization problem to minimize the cross-correlation integrated sidelobe level (CISL) for a MIMO sensor arrangement, constrained by unwrapped quadratic phase behavior. Simulation results confirm the effectiveness of the proposed approach.

This paper investigates channel-aware decision fusion empowered by massive MIMO systems and reconfigurable intelligent surfaces (RIS). By integrating both, we aim to improve goal-oriented (fusion) performance despite the unique propagation challenges introduced. Specifically, we investigate traditional favorable propagation properties in the context of RIS-aided Massive MIMO decision fusion. The above analysis is then leveraged (i) to design three sub-optimal simple fusion rules suited for the large-array regime and (ii) to devise an optimization criterion for RIS reflection coefficients based on long-term channel statistics. Simulation results confirm the appeal of the presented design.

This paper investigates the integration of Internet of Things (IoT) networks with modern massive multiple-input multiple-output (MIMO) wireless systems to enable various new use cases. Given the dynamic nature of parameters monitored by IoT nodes, efficient techniques for tracking these time-varying parameters are required. In a typical IoT networks, each IoT node linearly precodes its observations and transmits them over a coherent multiple access channel to a fusion center (FC). These IoT networks are power and bandwidth constrained in nature. Therefore, designing transmit precoders for the IoT nodes and a combiner for the FC is essential. This work proposes online linear receive combiner and transmit precoder designs that minimize the mean square error (MSE) under transmit power constraints. Using an alternating optimization technique, we derive closed-form solutions for the combiners and transmit precoders. Our numerical results validate the effectiveness of the proposed algorithms.

The evolution of array signal processing technologies is progressing toward the deployment of compact, densely arranged sensors to form extremely large aperture arrays (ELAA), aiming to significantly improve angular resolution and beamforming gain. In this paper, we propose a wideband near-field sensing system that combines orthogonal frequency-division multiplexing (OFDM) signaling with ELAA technology. The ELAA transmitter emits wideband signals, while the radar receiver processes the echoes to estimate critical target parameters, including location, velocity, and radar cross-section. We then derive the generalized Cramér-Rao lower bound (CRB) to assess the OFDM system’s estimation performance. Numerical experiments reveal that the proposed wideband near-field sensing system outperforms its far-field and narrowband counterparts by achieving lower CRB values.

A key challenge in Integrated Sensing and Communications (ISAC), especially in Full-Duplex (FD) Dual-Functional Radar-Communication (DFRC) systems, is self-interference (SI) caused by signal leakage from the transmitter to the receiver, impairing sensing tasks. Reconfigurable Intelligent Surface (RIS) can manipulate signal reflections with minimal power, making them promising for enhancing 6G communication, yet its potential for SI mitigation in DFRC systems is underexplored. This paper proposes a novel RIS-enabled approach for SI elimination in mono-static full-duplex DFRC systems. Our method decomposes the RIS function into spatial beam-forming and temporal modulation to ensure sufficient target illumination and orthogonality between reflected and transmitted waveforms, effectively eliminating SI without the need for SI channel information. Simulations confirm the effectiveness of the proposed approach in SI elimination and sensing performance.

In this paper, we investigate a novel computation resource allocation scheme for reconfigurable intelligent surfaces (RIS) backscatter communication (BackCom) enabled integrated sensing, communication and computation (ISCC) systems. We consider the joint design of transmit beamforming at the BS and the reflecting coefficients at the RIS as well as the computation resource allocation of each user. The optimization problem for the max computation efficiency (CE) under the constraints of power consumption, the Cramér-Rao bound (CRB) for angles estimation and communication requirement of each user is formulated. To deal with the intractable optimization problem, the alternative optimization (OA) and the alternating direction method of multipliers (ADMM) algorithm is developed. Furthermore, a more computationally efficient approach is introduced, which utilizes transmit beamforming based on an accelerated primal gradient (APG) method. Furthermore, the approximation principle is proposed to transform non-convex constraints in the optimization of the reflection coefficients at RISs. Simulation results show that introduction of RIS-BackCom can improve the efficiency of computing and maintain the tradeoff between CE and sensing performance.

Momentum-based optimizers are widely adopted for training neural networks. However, the optimal selection of momentum coefficients remains elusive. This uncertainty impedes a clear understanding of the role of momentum in stochastic gradient methods. In this paper, we present a frequency domain analysis framework that interprets the momentum method as a time-variant filter for gradients, where adjustments to momentum coefficients modify the filter characteristics. Our experiments support this perspective and provide a deeper understanding of the mechanism involved. Moreover, our analysis reveals the following significant findings: high-frequency gradient components are undesired in the late stages of training; preserving the original gradient in the early stages, and gradually amplifying low-frequency gradient components during training both enhance performance. Based on these insights, we propose Frequency Stochastic Gradient Descent with Momentum (FSGDM), a heuristic optimizer that dynamically adjusts the momentum filtering characteristic with an empirically effective dynamic magnitude response. Experimental results demonstrate the superiority of FSGDM over conventional momentum optimizers.

In dual-function radar-communication (DFRC) systems, precise and concurrent estimations of channel state information (CSI) and target directions are imperative to ensure simultaneous communications and sensing. This paper delves into a massive MIMO system employing hybrid beamforming (HBF), identified as a viable solution for achieving significant antenna gains while maintaining manageable hardware costs. The study focuses on a MIMO-DFRC system with a subarray-connection HBF architecture, proposing a preamble-by-preamble methodology to enable simultaneous wireless communications and target tracking on a per-frame basis. In the initial frames, we exploit the Doppler discrepancies between communication paths and fast-moving targets to segregate them. This sets the stage for iterative refinements of Doppler frequencies, angles of departure (AoDs), and angles of arrival (AoAs) estimations in the least square manner. Utilizing these estimations accrued from previous frames, the hybrid precoder and combiner of the subsequent frames are designed to boost a weighted signal-to-noise ratio (SNR), safeguarding the target tracking accuracy while concurrently refining the CSI estimation. Numerical simulations validate the proposed algorithm, demonstrating effective tracking performance with low overhead in MIMO-DFRC systems.

Hybrid analog-digital beamforming stands out as a key enabler for future communication systems with a massive number of antennas. In this paper, we investigate the hybrid precoder design problem for angle-of-departure (AoD) estimation, where we take into account the practical constraint on the limited resolution of phase shifters. Our goal is to design a radio-frequency (RF) precoder and a base-band (BB) precoder to estimate AoD of the user with high accuracy. To this end, we propose a two-step strategy where we first obtain the fully digital precoder that minimizes the angle error bound, and then the resulting digital precoder is decomposed into an RF precoder and a BB precoder, based on the alternating optimization and the alternating direction method of multipliers. Furthermore, we derive the quantization error upper bound and provide convergence conditions for the proposed algorithm. Numerical results demonstrate the superior performance of the proposed method over state-of-the-art baselines.

Target localization technology is widely applied in various applications, such as rescue missions, robot navigation, and the Internet of Things. However, in some scenarios, the positions of sensors are unknown due to the load limitation of the sensor carriers and environmental interferences, resulting in the instability of the global navigation satellite system (GNSS). This paper focuses on optimal angle-of-arrival (AOA) sensor placement using multiple position-unknown sensors for target localization accuracy improvement. To guarantee the uniqueness of the target coordinate, at least two anchors are needed. The anchors are some static benchmark objects in the environment with priori known positions. Firstly, a new optimization problem for AOA target localization accuracy improvement incorporating position-unknown sensors and anchors is formulated. Secondly, the optimal theoretical localization accuracies of the unknown sensors and target are derived by minimizing the trace of the Cramér-Rao lower bounds (CRLBs). Thirdly, a mixture optimization method, including a geometrical initialization and the new proposed simultaneous perturbation stochastic approximation and adaptive momentum estimation (SPSA-Adam) algebraic algorithm, is developed. Then, the correctness of the new theoretical findings and the effectiveness of the proposed sensor placement optimization method are verified by simulation examples.

Indoor sensing is challenging because of the multi-bounce effect, spherical wavefront, and spatial nonstationarity (SNS) of the near-field effect. This paper addresses radio-based environment sensing considering these issues. Specifically, graph theory (GT) is used to model the multi-bounce propagation of the near field. In this manner, indoor reflectors/scatterers are modeled as vertices in a propagation graph, the multi-bounce paths are modeled by the edges linking the vertices. Besides, the coupled multipath parameters in the near field, i.e., range and angles, are denoted directly by the coordinates of vertices. Then, the space-alternating generalized expectation-maximization (SAGE) algorithm is adapted to the proposed GT-based dictionary-aided multi-bounce SAGE (GM-SAGE), where the searching parameters including range and angle of departure/arrival (AoD/AoA) are transformed to the coordinates of vertices in the graph. To accelerate the two-bounce sensing, a recursive strategy is proposed. Furthermore, geometric information-based radio-sensing ambiguities are analyzed. The proposed algorithm is validated through a synthetic scattering channel and realistic ray tracing (RT) in a complex indoor office. The results demonstrate that the proposed GM-SAGE can deal with multi-bounce channels, where the one-bounce paths and near-field two-bounce paths are used to reconstruct the scatterers in the environment, the residual higher-bounce paths are identified and hence avoid ghost/mirror detection. Numerical simulations also show the influence of signal-noise ratio (SNR), grid size of vertices, aperture size, and near-far-field effects.

In this paper, we present a generic signal model applicable to various distributed radar setups, encompassing both phased array (PA) and MIMO radar configurations. We consider a range of waveform modulation methods, including TDM, BPM, DDM, and fast time CDM. We devise a GLRT based detector for scenarios where the interference consists of colored noise plus a signal in a low-rank subspace and prove that the designed detector is CFAR. We demonstrate that when the CPI time is similar for the systems, the PA radar system exhibits better detection performance than MIMO, irrespective of the waveform modulation approach adopted. However, if the CPI time of the PA system is divided to the number of transmit waveforms utilized in the MIMO radar case (to account for the time needed for a PA radar to scan all angles), then in the presence of non-uniform interference, MIMO techniques, except TDM, surpass the performance of PA. Conversely, in cases of uniform interference, the performance of both MIMO techniques and PA are equivalent.

Space debris detection and tracking, a key enabler for Space Situational Awareness (SSA), poses two inherent challenges: (1) small-sized targets (e.g., 1 − 10 cm) posing detection difficulties for conventional ground-based radars (GBRs) and optical measurements; (2) large number resulting in a costly tracking exercise. To address these, this work utilizes intersatellite link (ISL) in the emerging low earth orbit (LEO) constellations to opportunistically sense debris. The spatially dense-distributed debris is modeled as a cluster to reduce the number of quantities estimated. Using a stochastic geometry-based channel model, a nested expectationbased SAGE2is proposed, building on space-alternativegeneration-estimation-maximization (SAGE) to estimate the cluster-based channel parameters. Finally, the debris clusters are localized using the ISL forming a bistatic sensing setup. Simulation results validate the proposed approach and show the proposed SAGE2is faster than the conventional SAGE in clustered multipath channels.

This work considers an IoT network comprising of several IoT sensor nodes (SNs), a passive intelligent reflecting surface (IRS), and a fusion center (FC). Each IoT SN observes multiple physical phenomena, and transmits its observations to the FC for post processing. This necessitates the need for efficient preprocessing of each SN’s observations to combat wireless fading effects and optimize transmit power utilization. In this context, this paper presents a novel approach that jointly designs the transmit precoding matrix (TPM) for IoT SNs and optimizes the phase reflection matrix (PRM) for the IRS. The resulting non-convex optimization problem is tackled through an alternating optimization framework, where the individual TPM and PRM design subproblems are further addressed using the majorization minimization (MM) framework. Notably, the proposed solution yields closed-form expressions for TPM and PRM in each MM iteration, making it particularly suitable for low-cost IoT SNs. Numerical results demonstrate the efficacy of the proposed approach by showcasing significant enhancements in estimation performance compared to IoT networks lacking an IRS component.

The ambiguity function (AF) is a crucial tool in characterizing the range-angle response of a multiple-input multiple-output (MIMO) radar system, which is intricately influenced by the transmit waveforms, receiving filters and also antenna configurations. Notably, the role of antenna configurations is less explored compared to the well-studied areas of waveforms and filters. In this article, we incorporate antenna positions as an additional design parameter alongside waveforms and filters to optimize the AF in a specific range-angle bin. We employ the mainlobe-to-integrated-sidelobe-level-ratio (MISLR) as a quantitative metric to assess the performance. The resulting optimization problem is inherently nonconvex, encompassing binary and unimodular constraints. To address this challenge, we reformulate the problem, enabling an alternating optimization approach for antenna positions and waveforms. Each iteration involves solving a sequence of quadratic constrained quadratic programming problems for the binarily constrained antenna position optimization and updating the waveforms iteratively via an analytical expression. Our simulation results validate the effectiveness of the proposed method as it achieves higher MISLR with the same number of antennas compared to conventional approaches. Moreover, the optimized antenna configurations notably enhance the balance between angular ambiguity and resolution.

Beam-hopping (BH) technology, integral to multi-beam satellite systems, adapts beam activation to the variable communication demands of terrestrial users. The optimization of power allocation and beam illumination scheduling constitutes the core design challenge in BH systems, especially under the constraint on a limited number of simultaneously active beams due to restricted radio frequency chain availability. This paper proposes a two-stage BH design solution, which minimizes energy consumption in BH satellite communications while accommodating the heterogeneous demands of users. The first stage addresses the coupling variables of power and beam status by recasting the allocation and scheduling problem through a statistical lens, thus breaking down the intricate relationship between variables. To manage the resulting non-convex challenge, we propose an iterative method that capitalizes on the optimality conditions inherent to this problem. This method is designed to procure a statistically-informed solution that aligns with our reformulated interpretation. Subsequently, the second stage maps this solution into a concrete beam illumination schedule, employing binary quadratic programming techniques. A penalty-based iterative method is applied, ensuring convergence to a locally optimal solution. Through numerical simulations, the proposed framework has been validated for its efficacy in improving energy efficiency and accurately matching demands.

Dual-function radar-communications (DFRC) systems generally employ reconfigurable intelligent surface (RIS) as a reflector in the wireless media to enable non-line-of-sight (NLoS) sensing and communications. Different from RIS, reconfigurable holographic surface (RHS) are the surfaces with an embedded feed. These surfaces are deployed at the transceiver thereby leading to a lightweight design and greater control of the radiation amplitude. In this paper, we propose a novel frequency-selective RIS-assisted wideband DFRC system that is also equipped with a RHS at the transceiver. Our goal is to jointly design the digital, holographic, and passive beam-formers to maximize the radar signal-to-interference-plus-noise ratio (SINR) while ensuring the communication SINR among all users. The resulting nonconvex optimization problem involves maximin objective and difference of convex constraint. We develop an alternating maximization framework to decouple and iteratively solve these subproblems. Numerical experiments demonstrate that the proposed method achieves better radar performance than non-RHS, non-RIS, and randomly-configured RIS-aided DFRC systems.

In the symbol-level precoding (SLP) based wireless systems, the reconfigurable intelligent surface (RIS) is usually configured on a block level, which causes a mismatch to the SLP design in terms of update rate. Although it is expected that updating both the RIS and precoding on the symbol level could boost the system performance, the requirements for synchronization and system overhead will become demanding inevitably. In this paper, we consider the RIS-aided dual-function radar-communication (DFRC) system and investigate the benefit of increasing the RIS updating frequency. We jointly design the SLP and RIS to maximize the target illumination power while satisfying the power budget and the multiuser multiple input single output (MU-MISO) communication quality of service (QoS) requirements, where the RIS is updated multiple times in a block (at a sub-block level). Through the simulation results, we demonstrate the optimized trade-off between system performance and RIS update rate.

Intelligent reflecting surface (IRS) is recognized as an enabler of future dual-function radar-communications (DFRC) by improving spectral efficiency, coverage, parameter estimation, and interference suppression. Prior studies on IRS-aided DFRC focus either on narrowband processing, single-IRS deployment, static targets, non-clutter scenario, or on the under-utilized line-of-sight (LoS) and non-line-of-sight (NLoS) paths. In this paper, we address the aforementioned shortcomings by optimizing a wideband DFRC system comprising multiple IRSs and a dual-function base station that jointly processes the LoS and NLoS wideband multi-carrier signals to improve both the communications SINR and the radar SINR in the presence of a moving target and clutter. We formulate the transmit, receive and IRS beamformer design as the maximization of the worst-case radar signal-to-interference-plus-noise ratio (SINR) subject to transmit power and communications SINR. We tackle this nonconvex problem under the alternating optimization framework, where the subproblems are solved by a combination of Dinkelbach algorithm, consensus alternating direction method of multipliers, and Riemannian steepest decent. Our numerical experiments show that the proposed multi-IRS-aided wideband DFRC provides over 4 dB radar SINR and 31.7% improvement in target detection over a single-IRS system.

Beam Hopping (BH) is a popular technique considered for next-generation multi-beam satellite communication system which allows a satellite focusing its resources on where they are needed by selectively illuminating beams. While beam illumination plan can be adjusted according to its needs, the main limitation of convectional BH is the adjacent beam avoidance requirement needed to maintain acceptable levels of interference. With the recent maturity of precoding technique, a natural way forward is to consider a dynamic beam illumination scheme with selective precoding, where large areas with high-demand can be covered by multiple active precoded beams. In this paper, we mathematically model such beam illumination design problem employing an interference-based penalty function whose goal is to avoid precoding whenever possible subject to beam demand satisfaction constraints. The problem can be written as a binary quadratic programming (BQP). Next, two convexification frameworks are considered namely: (i) A Semi-Definition Programming (SDP) approach particularly targeting BQP type of problems, and (ii) Multiplier Penalty and Majorization-Minimization (MPMM) based method which guarantees to converge to a local optimum. Finally, a greedy algorithm is proposed to alleviate complexity with minimal impact on the final performance. Supporting results based on numerical simulations show that the proposed schemes outperform the relevant benchmarks in terms of demand matching performance while minimizing the use of precoding.

In millimeter-wave (mmWave) dual-function radar-communication (DFRC) systems, hybrid beamforming (HBF) is recognized as a promising technique utilizing a limited number of radio frequency chains. In this work, in the presence of extended target and clutters, a HBF design based on the subarray connection architecture is proposed for a multiple-input multiple-output (MIMO) DFRC system. In this HBF, the double-phase-shifter (DPS) structure is embedded to further increase the design flexibility. We derive the communication spectral efficiency (SE) and radar signal-to-interference-plus-noise-ratio (SINR) with respect to the transmit HBF and radar receiver, and formulate the HBF design problem as the SE maximization subjecting to the radar SINR and power constraints. To solve the formulated nonconvex problem, the joinT Hybrid bEamforming and Radar rEceiver OptimizatioN (THEREON) is proposed, in which the radar receiver is optimized via the generalized eigenvalue decomposition, and the transmit HBF is updated with low complexity in a parallel manner using the consensus alternating direction method of multipliers (consensus-ADMM). Furthermore, we extend the proposed method to the multi-user multiple-input single-output (MU-MISO) scenario. Numerical simulations demonstrate the efficacy of the proposed algorithm and show that the solution provides a good trade-off between number of phase shifters and performance gain of the DPS HBF.

Direction-of-arrival (DOA) estimation can be represented as a sparse signal recovery problem and effectively solved by sparse Bayesian learning (SBL). For the DOA estimation in active sensing, the SBL-based estimation error is related to the transmitted probing waveform. Therefore, it is expected to improve the estimation by waveform optimization. In this paper, we propose a recurrent scheme of waveform design by sequentially leveraging on the previous-round SBL estimates. Within this scheme, we formulate the waveform design problem as a minimization of the SBL estimation variance, which is non-convex and then solved by a majorization-minimization based algorithm. The simulations demonstrate the efficacy of the proposed design scheme in terms of avoiding incorrect detection and accelerating the DOA estimation convergence. Further, the results indicate that the waveform design is essentially a beampattern shaping methodology.

Pulse compression can enhance both the performance in range resolution and sensitivity for weather radar. However, it will introduce the issue of high sidelobes if not delicately implemented. Motivated by this fact, we focus on the pulse compression design for weather radar in this paper. Specifically, we jointly design both the subpulse codes and extended mismatch filter based on the alternating direction method of multipliers (ADMM). This joint design will yield a pulse compression with low sidelobes, which equivalently implies a high signal-to-interference-plus-noise ratio (SINR) and a low estimation error on meteorological reflectivity. The experiment results demonstrate the efficacy of the proposed pulse compression strategy since its achieved meteorological reflectivity estimations are highly similar to the ground truth.

This paper deals with the problem of the hybrid beamforming design of wideband orthogonal frequency division multiplexing (OFDM) dual-function radar-communication (DFRC) system, which is expected to achieve a satisfactory user's spectral efficiency and form excellent space-frequency spectrum behavior as well as spatial nulls on the directions of strong signal-dependent interference (such as clutters) simultaneously. For such purpose, we formulate our problem by maximizing the communication spectral efficiency subject to the constraints of radar integrated sidelobe to mainlobe ratio (ISMR) and spatial spectrum nulling (SSN). Due to the fact that the analog beamformer for all subcarriers and digital beamformer for each subcarrier are simultaneously optimized in the wideband OFDM system, the resultant problem is difficult to solve. Towards that end, an efficient algorithm is devised based on the consensus alternating direction method of multipliers (CADMM) framework. Numerical simulation results demonstrate the superiority of the proposed hybrid beamforming algorithm.

Abstract The joint design problem for the coexistence of multiple‐input multiple‐output (MIMO) radar and multi‐user multiple‐input‐single‐output (MU‐MISO) communication is investigated. Different from the conventional design schemes, which require defining the primary function, we consider designing the transmit waveform, precoding matrix and receive filter to maximize the radar SINR and the minimal SINR of communication users, simultaneously. By doing so, the promising overall performance for both sensing and communication is achieved without requiring parameter tuning for the threshold of communication or radar. However, the resulting optimization problem which contains the maximin objective function and the unit sphere constraint, is highly nonconvex and hence difficult to attain the optimal solution directly. Towards this end, the epigraph‐form reformulation is first adopted, and then an alternating maximisation (AM) method is devised, in which the Dinkelbach’s algorithm is used to tackle the nonconvex fractional‐programing subproblem. Simulation results indicate that the proposed method can achieve improved performance compared with the benchmarks.

Optimal allocation of shared resources is key to deliver the promise of jointly operating radar and communications systems. In this paper, unlike prior works which examine synergistic access to resources in colocated joint radar-communications or among identical systems, we investigate this problem for a distributed system comprising heterogeneous radars and multi-tier communications. In particular, we focus on resource allocation in the context of multi-target tracking (MTT) while maintaining stable communications connections. By simultaneously allocating the available power, dwell time and shared bandwidth, we improve the MTT performance under a Bayesian tracking framework and guarantee the communications throughput. Our${a}$lter${n}$ating allo${c}$ation of${h}$eterogene${o}$us${r}$esources (ANCHOR) approach solves the resulting non-convex problem based on the alternating optimization method that monotonically improves the Bayesian Cramér-Rao bound. Numerical experiments demonstrate that ANCHOR significantly improves the tracking error over two baseline allocations and stability under different target scenarios and radar-communications network distributions.

Beampattern synthesis is a key problem in many wireless applications. With the increasing scale of MIMO antenna array, it is highly desired to conduct beampattern synthesis on a sparse array to reduce the power and hardware cost. In this paper, we consider conducting beampattern synthesis and sparse array construction jointly. In the formulated problem, the beam-pattern synthesis is designed by minimizing the matching error to the beampattern template, and the Shannon entropy function is first introduced to impose the sparsity of the array. Then, for this nonconvex problem, an iterative method is proposed by leveraging on the alternating direction multiplier method (ADMM) and the majorization minimization (MM). Simulation results demonstrate that, compared with the benchmark, our approach achieves a good trade-off between array sparsity and beampattern matching error with less runtime.

Long-range radio-frequency wireless charging has received more and more attention in recent years. Unlike the literature that focuses on the mobile charging scenario, which is formulated as a path-planning problem, we consider a static network that consists of a power beacon (PB) and multiple wireless powered user nodes. Each user node is equipped with a rechargeable battery and harvests energy only from the PB. We assume that all user nodes are either in charge or work mode. Under this assumption, we first propose a charge scheduling scheme that achieves the system's maximal energy efficiency. We then further investigate the system's improvement with a multiantenna PB using the energy beamforming technique. Next, we extend our scheduling scheme to a two-tier network architecture, where a first-tier PB first transfers energy to the second-tier sub-PBs, and then those sub-PBs deliver energy to the user nodes which belong to their clusters. It is shown that energy beamforming with multiple antennas brings a significant improvement to the system's performance. Moreover, the two-tier architecture is shown to be superior to the one-tier architecture in terms of the energy transfer efficiency and the system's realization complexity. Finally, the simulation results demonstrate our theoretical findings.