arXiv subjects

Qingqing Wu

Publications and source records attributed to Qingqing Wu.

At least 19 recordsLinked to original sources

arXiv2026

Modality-Decoupled Federated Learning for Privacy-Preserving Embodied Intelligence in 6G

Sixth-generation (6G) wireless networks are expected to provide a key infrastructure for large-scale embodied intelligence, where heterogeneous robots collaborate through low-latency connectivity, edge intelligence, and distributed sensing. Vision-language-action (VLA) models offer a foundation by integrating visual perception, language understanding, and action generation into a unified closed-loop policy. However, training and adapting VLA models to distributed robotic agents introduce challenges in privacy protection, communication efficiency, and model heterogeneity. Existing federated learning (FL) methods overlook the intrinsic differences among vision, language, and action pathways in parameter scale, privacy exposure, update dynamics, and tolerance to compression or perturbation. To address this issue, this article proposes FedMVLA, a modality-decoupled FL framework for privacy-preserving embodied intelligence in 6G networks. FedMVLA incorporates three mechanisms: modality-aware federated aggregation (MAFA), modality-aware privacy allocation (MAPA), and modality-aware communication compression (MACO), together with a modality-sliced transport design that routes the precision-critical action stream through a protected ultra-reliable low-latency slice. A case study on federated robotic manipulation over the Third Generation Partnership Project (3GPP)-based wireless substrate, covering fading, co-channel interference, and malicious jamming, shows that FedMVLA achieves an 84.8% task success rate, exceeds FedAvg by 22.2 percentage points, sustains a widening margin when scaling to 128 clients across eight cells, and reduces the schedule-averaged per-client uplink model-update payload by 95.6% (approximately 96%), while keeping the 95th percentile (p95) of the round-critical uplink completion time near 1.5s.

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arXiv2026

Rotatable Antenna Enabled Multi-Satellite Communications: Joint Satellite Selection and Boresight Trajectory Optimization

This paper considers a satellite-to-ground communication system in which a ground station (GS) equipped with independently rotatable antenna (RA) elements jointly decodes independent streams from multiple low-Earth-orbit (LEO) satellites over a shared time--frequency resource. Specifically, we formulate a two-timescale throughput maximization problem under exogenous cochannel interference, capturing serving-set composition, RA-enabled channel shaping, time-varying satellite geometry, and mechanically constrained inter-epoch reconfiguration. We first characterize the joint effects of interference-whitened channel strength and spatial separability on multi-satellite reception, motivating the joint design of satellite selection and RA control. With the RA trajectory fixed, we establish the monotone submodularity of the epoch-level selection objective and construct an incumbent-tight modular lower-bound surrogate, leading to an efficient discrete Minorization-Maximization (MM) selection algorithm. For fixed serving sets, we develop slew-feasible RA updates based on Riemannian gradients and organize them into a two-color parallel update scheme. The two blocks are integrated into a monotone alternating algorithm with guaranteed objective convergence. Simulations demonstrate consistent gains over benchmark schemes and reveal an optimal balance between channel strength and spatial separability. The results further show that satellite selection is particularly important in underloaded and actuator-limited regimes, whereas RA shaping becomes more influential near full spatial loading.

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arXiv2026

Intelligent Reflecting Surface Deployment for Low-Altitude Coverage: Illumination Geometry, Directional Characteristics, and Optimization

Terrestrial base stations (BSs) are typically configured with fixed downtilt to serve ground users, resulting in weak illumination of low-altitude airspace even under line-of-sight (LoS) propagation. In this paper, we establish a channel model that incorporates BS and intelligent reflecting surface (IRS) radiation patterns for three-dimensional (3D) low-altitude coverage while preserving the existing BS configuration. We formulate a budget-constrained IRS deployment problem that jointly determines candidate-site selection, IRS orientations, and phase shifts to maximize the worst-case signal-to-noise ratio (SNR) over the 3D low-altitude airspace. The selected sites and optimized IRS parameters remain fixed after deployment, yielding a quasi-static IRS configuration. We characterize the illumination geometry between the fixed-downtilt BS and rooftop candidates by deriving the nonnegative installation-height range satisfying the BS main-lobe condition. The separation between the mapped main-lobe height boundaries grows linearly with horizontal BS-to-site distance and decreases inversely with the number of BS antennas. We further derive an analytical lower bound on the regional worst-case normalized array gain achievable through IRS phase design over served directions with different direction spans. The resulting sufficient direction span decreases inversely with the square root of the number of IRS elements when the same worst-case normalized gain guarantee is maintained. We develop a mixed-integer alternating optimization (AO) algorithm to solve the resulting problem. Simulation results validate the analytical characterizations and show that the proposed scheme achieves higher worst-case SNR than benchmarks across different deployment budgets.

cs.IT↗

arXiv2026

Rotatable Antenna Relaying: Joint Precoding and Antenna Pointing Design

This paper investigates a rotatable antenna (RA)-enhanced half-duplex amplify-and-forward relaying system, where a multi-antenna base station (BS) serves multiple single-antenna users via a multi-antenna relay. The BS and users employ isotropic antennas, whereas the relay employs directional RAs whose pointing matrix is shared by both hops to avoid inter-hop reorientation delay and control overhead. We aim to maximize the minimum signal-to-interference-plus-noise ratio (SINR) among all users by jointly optimizing the BS precoding, relay precoding, and RA pointing matrices. To tackle this non-convex problem, we first investigate the single-user scenario and reduce the joint design to RA pointing optimization using the optimal relay precoding matrix available in closed form. A manifold-aware Frank--Wolfe (MFW) method is then employed to obtain a suboptimal pointing solution. Under a symmetric far-field line-of-sight geometry, we further characterize the globally optimal pointing structure and derive a directivity threshold separating common pointing from antenna splitting. Building on this MFW procedure, we next address the general multiuser scenario. Specifically, the quadratic transform is first applied to obtain an equivalent auxiliary-variable formulation, which is subsequently solved suboptimally via alternating optimization (AO). In particular, we employ a safeguarded extension of the MFW method based on log-sum-exp smoothing to update the RA pointing matrix in each AO iteration. Simulation results demonstrate that the proposed algorithms consistently achieve the best signal-to-noise ratio (SNR) and SINR performance among all considered schemes. It is further shown that the optimized RA pointing adapts to the two-hop geometry, the preferred directivity factor depends on user load, and relay placement with relatively balanced two-hop propagation conditions is generally preferable.

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arXiv2026

Sparse Rotatable Arrays (SRA): Unifying Array Aperture and Antenna Directivity for Wireless Communications

Sparse rotatable array (SRA) is a novel reconfigurable antenna architecture that jointly exploits sparse aperture configuration and antenna directivity to enhance spatial resolution for future wireless communications. Specifically, SRA activates a subset of rotatable antennas over a large candidate aperture and adjusts their boresight directions, thereby creating a directionally selective sparse aperture with reduced hardware requirements and enhanced spatial flexibility. In this paper, we investigate an SRA-aided multi-group communication system, where users are organized into spatial groups with different service requirements. We develop a group-aware SRA design framework by jointly optimizing the sparse-aperture allocation, RA orientations, and transmit beamforming to maximize the weighted max-min signal-to-interference-plus-noise ratio (SINR). Then, we characterize the operating principles of SRA and reveal that sparse aperture improves spatial resolution by enlarging the effective array aperture, while antenna directivity suppresses inter-group coupling through directional control, thereby enabling simplified group-wise beamforming structures. Guided by these insights, we develop a structured low-complexity alternating optimization algorithm that embeds a closed-form projected group-center RA orientation rule into the sparse-aperture allocation and beamforming design. The proposed algorithm combines analysis-guided initialization, sampled multi-start antenna-allocation search, and bisection-based second-order cone programming for beamforming. Numerical results show that the proposed SRA design closely approaches fully-shared SRA benchmarks and significantly outperforms compact subarray and omni sparse-array schemes.

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arXiv2026

Mobile Tracking via Target-Mounted IRS-Assisted ISAC System

This paper proposes a target-mounted intelligent reflecting surface (IRS)-assisted integrated sensing and communication framework for real-time unmanned aerial vehicle (UAV) tracking, addressing challenges such as link blockage and weak radar cross section in the low-altitude economy. By integrating the IRS onto the UAV, the system creates a mobile cooperative target that provides controllable line-of-sight echoes for self-tracking while acting as a mobile relay for ground communication enhancement. We establish a comprehensive three dimensions state evolution model for the maneuvering UAV. Based on this model, an extended Kalman filter is immediately implemented to achieve real time tracking of the moving UAV. To characterize the fundamental theoretical limits of this recursive estimation process, we derive the analytical posterior Cramer Rao bound and a closed form expression for the elliptical tradeoff performance bound to quantify the relationship between sensing precision and communication throughput. To ensure millisecond level responsiveness, we develop a low complexity joint beamforming design. By utilizing the analytical mapping between tracking and communication requirements, the proposed scheme yields closed form solutions for beamforming vectors, effectively bypassing the time consuming numerical iterations of conventional methods. Numerical simulations demonstrate that the proposed framework significantly outperforms traditional fixed-deployment benchmarks across complex maneuvering trajectories, achieving centimeter-level accuracy while substantially reducing transmit power and processing latency.

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arXiv2026

Convergence-Latency-Aware Adaptive Modulation and Resource Allocation in RIS-Assisted Wireless Federated Learning

Federated learning (FL) over wireless networks suffers from significant training latency and degraded convergence due to unreliable wireless transmission, especially under blocked propagation environments. Although reconfigurable intelligent surfaces (RISs) can improve communication reliability, existing wireless FL studies rarely characterize the trade-off between learning convergence and communication delay under modulation-dependent transmission errors. In this paper, we consider a wireless FL system operating under RIS-assisted blocked-link propagation scenarios, and focus on adaptive modulation and sub-channel allocation for convergence-latency aware communication design. By characterizing the effect of symbol errors on uploaded local gradients, we derive a convergence-related upper bound that reveals the impact of symbol error rate (SER) on FL loss decay. Based on this result, we formulate a joint convergence-latency optimization problem, which is cast as a mixed-integer nonlinear programming (MINLP) problem, and solve it using a low-complexity hybrid alternating optimization framework. Extensive experiments on MNIST, CIFAR-10, and Speech Commands show that the proposed scheme consistently achieves faster convergence and higher test accuracy than existing adaptive communication schemes, especially in complex tasks and challenging wireless scenarios.

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arXiv2026

Beyond Beamforming: Phase-and-Gain Channel Shaping via Rotatable Antenna Arrays

This paper investigates geometry-reconfigurable transmission for multiuser communication systems enabled by a rotatable antenna array. In contrast to conventional fixed arrays, the proposed architecture jointly exploits array pose adjustment and element-level boresight steering, thereby reshaping both the array-induced phase responses and the direction-dependent channel gains. We formulate a weighted sum-rate maximization problem that jointly optimizes the transmit beamformers, array pose, and element boresights under practical visibility and steering constraints. To reveal the underlying design principles, we first provide a geometric interpretation via zero-forcing analysis, showing that the resulting rates stem from both channel-strength enhancement and spatial-separability improvement. Specifically, array-pose rotation improves inter-user channel orthogonality even with isotropic elements, whereas directional elements introduce a tradeoff between phase-based spatial separation and boresight-dependent gain alignment. Motivated by these insights, we develop an efficient optimization framework that jointly coordinates transmit beamforming, array-pose adaptation, and element-boresight steering to exploit the geometry-induced phase-and-gain channel-shaping capability. Simulation results demonstrate that the proposed joint design outperforms fixed-array, pose-only, and boresight-only benchmarks, with larger gains achieved under more directive element patterns and tighter boresight-steering constraints.

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arXiv2026

Ultra-Low-Cost Hybrid Beamforming: A New Static-Connection Architecture with Sparse Phase-Shifter Sharing

Hybrid beamforming is a promising solution for high-frequency multi-antenna wireless systems, but its implementation is constrained by the cost and complexity of analog phase-shifter (PS) networks. Although sub-connected architectures simplify the analog network, their conventional realization still requires a dedicated PS for each antenna, causing considerable layout area, wiring, calibration, and control overheads. To address this issue, this paper proposes a novel static-connection architecture with sparse PSs for ultra-low-cost sub-connected hybrid beamforming, where antennas within each sub-array share a PS through an optimized fixed PS-to-antenna connection matrix. The proposed architecture preserves static connections while enabling dynamic beam control via adaptive PS phase-shift adjustments and digital precoding. For the single-radio-frequency (RF)-chain scenario, the sparse-PS connection design is transformed into an antenna-grouping problem, with analytically characterized structural properties and an efficient algorithm. For the multi-RF-chain scenario, we develop a quality-of-service (QoS)-majorization-minimization (MM) algorithm to handle the mixed discrete-continuous optimization problem. Numerical results demonstrate that the proposed architecture reduces the PS count while preserving most beamforming capability of the traditional full-PS sub-connected architecture. In particular, the proposed design achieves PS-count reductions of 37.5% and 62.5% in single-RF-chain and multi-RF-chain systems, respectively, while avoiding deep-null and grating-lobe degradations associated with deterministic connection schemes. These results provide engineering insights into static sparse-PS sharing: the key to hardware-efficient hybrid beamforming is not merely reducing the PS count, but also preserving essential analog-domain degrees of freedom through optimized PS connection topologies.

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arXiv2026

Sensing for Reliable UAV Communication: Robust Trajectory and Resource Optimization in Low-Altitude Networks

In low-altitude wireless networks, sensing-aided communication has emerged as a promising integrated sensing and communication (ISAC) paradigm for unmanned aerial vehicle (UAV) tracking and communication. This paper investigates reliable sensing-aided communication for multiple cellular-connected UAVs under mobility uncertainties. Specifically, we maximize the minimum outage capacity among UAVs by jointly optimizing their real-time predicted positions, as well as the base station (BS) transmit power and bandwidth allocations. To address the non-convex and intractable maximum tolerable outage probability (OP) constraints, two robust optimization schemes are proposed based on a continuous confidence ellipse (CE) and discretized inverse-whitened sectors (IWSs), respectively. For the CE-based scheme, an efficient algorithm is proposed to optimize the predicted UAV positions individually via block successive convex approximation, followed by convex resource allocation. For the IWS-based scheme, an IWS-based OP approximation is proposed to facilitate the robust optimization, based on which a low-complexity IWS selection method is proposed to decouple the optimization variables. Then, a similar sequential optimization algorithm is proposed based on the projected gradient descent approach. The two algorithms are further unified into a common trajectory-resource optimization framework, revealing a low-complexity structure for robust UAV trajectory and resource management. Simulation results validate the effectiveness of our proposed OP approximation, demonstrate the significant outage capacity improvement of the proposed robust optimization schemes over benchmark schemes, and illustrate the superiority of the IWS-based scheme over the CE-based scheme.

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arXiv2026

Active Sensing-assisted UAV Communications with Jittering: Framework and Performance Analysis

Providing reliable communication for unmanned aerial vehicles (UAVs) via existing cellular networks is crucial for enabling the rapid growth of the low-altitude economy. However, UAV jittering significantly degrades communication quality due to induced beam misalignment. Inspired by recent advances in integrated sensing and communication, we propose a novel two-stage active sensing-assisted communication framework tailored for ground-to-UAV links with jittering. Specifically, two schemes are conceived to leverage sensing for enhancing communication performance, namely the communication-oriented scheme and the sensing-oriented scheme. For the sensing-oriented scheme, deterministic signals are employed in the first stage to facilitate angle-of-arrival (AoA) acquisition at the UAV side, followed by pure communication service in the second stage by using the estimated AoA. In contrast, the communication-oriented scheme employs Gaussian information-bearing signals throughout both stages, with AoA estimation relying on Gaussian random signals. For both schemes, we provide maximum likelihood estimators for AoA, along with analytical results characterizing the Cram\'er-Rao bound. To capture the performance limit, closed-form expressions for the achievable rates of the two schemes are derived, unveiling a fundamental tradeoff between sensing and communication quality across the two stages by tuning the time allocated to the first stage. The optimal time allocation that maximizes the overall rate is obtained in semi-closed-form. Based on these results, we unveil a sufficient condition under which the communication-oriented scheme outperforms the sensing-oriented scheme, which admits an interesting threshold-based structure. Asymptotic analysis demonstrates that the performance loss of the proposed schemes relative to the jitter-free upper bound approaches zero in the high transmit power regime.

cs.IT↗

arXiv2026

Fundamentals of NOMA in Low-Earth Orbit Coordinated Multi-Satellite Networks

Coordinated multi-satellite (CoMS) transmission and non-orthogonal multiple access (NOMA) are envisioned to jointly enhance coverage, capacity, and spectrum efficiency for satellite networks. Their integration into a unified CoMS-NOMA framework will allow more efficient, reliable, and energy-efficient multi-user access. This paper investigates the downlink performance of CoMS-NOMA networks from a system-level perspective, in which multiple satellites cooperatively serve multiple users via NOMA. Leveraging tools from stochastic geometry, related angles and distances in CoMS-NOMA are first derived as intermediate results. Then, we obtain the combined signal power distributions and analyze coverage and spectrum performance under both inter- and intra-satellite interference, accounting for potential imperfect successive interference cancellation (SIC). The analytical model is validated across a range of system parameters, including the number of satellites, service region angle, error-propagation factor, and power allocation coefficients. Numerical results indicate that increasing the number of cooperative satellites does not always improve coverage and spectrum efficiency. Additionally, while a higher main-lobe gain improves coverage, a near-perfect SIC provides only slightly greater benefits than a reasonably good SIC. With properly selected power allocation coefficients, CoMS-NOMA achieves up to a 270% improvement in coverage and a 56% gain in sum spectral efficiency, compared with conventional orthogonal and single-satellite schemes, indicating potential for green, energy-efficient satellite networking.

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arXiv2026

Multi-User MIMO with Rotatable Antennas and IRS: Joint Antenna Boresight and IRS Orientation Design

In this paper, we investigate an intelligent reflecting surface (IRS)-assisted multi-user system, where the base station (BS) employs rotatable antennas (RAs) and the IRS can adjust the panel orientation.To alleviate the severe multiplicative path loss of the cascaded channel, the IRS is deployed near the BS, while the user-BS and user-IRS links remain in the far field. We formulate a sum-rate maximization problem by jointly optimizing the receive beamforming, IRS phase shifts, BS antenna boresights, and IRS panel orientation. To tackle the resulting highly coupled and non-convex problem, we first study a single-user case to reveal the structure of the dual-rotation gain, which is shown to be multiplicatively separable in the far field but coupled in the near field. For the general multi-user case, we develop an alternating optimization algorithm, where the receive beamforming is updated in closed form, the IRS phase shifts are optimized by an FP-assisted Riemannian conjugate gradient method, and the BS antenna boresights and IRS panel orientation are updated via projected gradient methods. Simulation results demonstrate the significant sum-rate gains achieved by the proposed coordinated rotation design over fixed-orientation and single-rotation benchmark schemes, and provide useful insights into near-field dual-rotation design.

cs.IT↗

arXiv2026

Rotatable Antenna-Enhanced Wireless Sensing with Uniform Sparse Array via Tensor Decomposition

In this letter, we propose a new wireless sensing system equipped with a rotatable antenna (RA) array to enhance the sensing performance of a uniform sparse array (USA). To tackle the severe spatial undersampling issues, we propose a novel tensor decomposition-based direction-of-arrival (DOA) estimation algorithm. Specifically, we introduce a synchronous multiple rotation pattern for active target probing such that the received signals across multiple rotations to capture the diverse spatial degree of freedoms. Subsequently, we mathematically formulate the received signals across successive rotations as a third-order tensor, and leverage the canonical polyadic decomposition to obtain the factor matrices incorporating the DOA of targets. By analyzing the extrema distribution laws of array steering vector correlation (SVC) and gain SVC of RAs, we propose to combine the array and gain factor matrices via the Kronecker product, which theoretically guarantees the unambiguous DOA estimation. Simulation results demonstrate that the proposed RA-enhanced tensor decomposition-based algorithm achieves high-precision and unambiguous sensing performance compared to conventional uniform dense arrays and omnidirectional antenna systems.

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arXiv2026

Joint Transmit and Receive Antenna Orientation Design for Secure MIMO Communications

Physical layer security (PLS) is a promising paradigm for safeguarding 6G wireless networks by exploiting the inherent characteristics of wireless channels. However, the efficiency of conventional PLS is often limited by fixed orientation antennas. This paper investigates a rotatable antenna (RA)-aided secure multiple-input multiple-output (MIMO) communication system, where both the transmitter and the receiver are equipped with RAs in the presence of an eavesdropper. By dynamically optimizing the orientations of RAs, we can proactively reshape the effective MIMO channels to enhance legitimate transmission while simultaneously suppressing information leakage to the eavesdropper. We formulate a secrecy rate maximization problem by jointly optimizing the transmit beamforming, artificial noise (AN) covariance matrix, and the transmit/receive RA orientations, subject to the transmit power budget and antenna orientation constraints. To tackle the resulting highly coupled and non-convex problem, we first study a simplified single-input single-output (SISO) case to reveal the structure of the optimal RA orientation. For the general MIMO case, we develop an alternating optimization algorithm by reformulating the original problem through the minimum mean-square error framework. In particular, the transmit beamforming and AN covariance matrix are derived in semi-closed forms, while the RA orientations are updated via the Riemannian Frank-Wolfe method. The proposed design is further extended to the multi-receiver secure transmission scenario. Simulation results show that the proposed scheme converges rapidly and achieves significant secrecy rate gains over the conventional fixed-orientation scheme.

cs.IT↗

arXiv2026

Two-Timescale Design for Rotatable-Antenna Systems With Imperfect CSI: Rate Analysis and Orientation Optimization

This paper studies uplink multiuser MIMO with a rotatable antenna (RA) array under imperfect channel state information (CSI), where each base-station antenna can adjust its boresight direction within an angular region. To balance performance and control overhead, we propose a two-timescale design: RA orientations are optimized from statistical CSI on a large timescale, while linear receive combiners are updated per coherence block from linear minimum-mean-squared-error (LMMSE) channel estimates. Under this framework, we derive a closed-form use-and-then-forget (UatF)-based rate expression for maximum-ratio combining (MRC) and a closed-form statistical rate surrogate for weighted zero-forcing (wZF) under imperfect CSI, revealing how RA rotation influences useful signal strength, estimation-error-induced self-interference, and multiuser interference. The analysis shows that the orientation minimizing channel-estimation error differs from the rate-maximizing one, and that MRC and wZF prefer different rotation configurations due to their distinct mechanisms of signal aggregation and error-aware user separation. For the resulting non-convex rotation design problems, we develop a projected-gradient algorithm over a product of spherical caps with explicit derivatives of the required channel statistics and rate metrics. Numerical results verify the accuracy of the large-timescale surrogates and show substantial performance gains from RA optimization.

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arXiv2026

Sensing-Assisted Secure Communication in MA-Aided ISAC: CRB Analysis and Robust Design

A core challenge in physical-layer security is the difficulty of obtaining the channel state information (CSI) of potential eavesdroppers. The inherent sensing functionality of integrated sensing and communication (ISAC) systems offers a promising solution by enabling the estimation of key parameters, such as the eavesdropper's angles of departure (AoDs). Capitalizing on this capability, we propose a sensing-assisted secure communication scheme for a movable antenna (MA)-aided ISAC system. The scheme comprises two stages: eavesdropper AoD sensing and secure communication. In the first stage, the base station (BS) optimizes the positions of its transmit and receive MAs to enhance sensing accuracy. We derive the closed-form Cramer-Rao bound (CRB) for the estimated AoDs to fundamentally characterize how MA positions influence the estimation uncertainty. In the second stage, the BS ensures secure communication by designing a robust beamforming vector that accounts for the AoD uncertainty region and by further optimizing the transmit MAs' positions to maximize the secrecy rate. To manage the end-to-end design, we formulate a joint optimization problem. This intractable non-convex problem is decomposed into two subproblems. For the first subproblem, we develop an alternating optimization (AO) algorithm to solve the CRB minimization problem. For the second subproblem, we solve the worst-case secrecy rate maximization problem using a method based on backward induction, convex hull construction, and AO. Finally, simulation results are provided to demonstrate the significant advantages of the proposed scheme compared to various benchmarks.

cs.IT↗

arXiv2026

Trajectory Design for Fairness Enhancement in Movable Antennas-Aided Communications

Through adaptive antenna repositioning, the movable antenna (MA) technology enables on-demand reconfiguration of wireless channels, thereby creating an additional spatial degree of freedom in improving communication performance. This paper investigates a multiuser uplink communication system aided by MAs, where a base station (BS) equipped with multiple MAs serves multiple single-antenna users. Specifically, given that an optimized array geometry cannot guarantee rate fairness, we focus on designing antenna trajectory at the BS to maximize the minimum achievable rate among all users over a finite time period. The resulting optimization problem is fundamentally challenging to solve due to the continuous-time nature. To address it, we first examine an ideal case with infinitely fast MA movement and demonstrate that the relaxed problem can be optimally solved via the Lagrangian dual method. The obtained trajectory solution reveals that the BS should employ a finite set of MA deployment patterns, each allocated an optimal time duration. Building on this, we then study the general case with limited MA movement speed and propose a heuristic trajectory design inspired by the optimal patterns identified in the ideal scenario. Several insights are also gained by examining the simplified special case. Finally, numerical results are provided to validate the effectiveness of the proposed designs compared to competitive benchmarks.

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