Low-Complexity Algorithm for Stackelberg Prediction Games with Global Optimality

Stackelberg prediction games (SPGs) model strategic data manipulation in adversarial learning via a leader--follower interaction between a learner and a self-interested data provider, leading to challenging bilevel optimization problems. Focusing on the least-squares setting (SPG-LS), recent work shows that the bilevel program admits an equivalent spherically constrained least-squares (SCLS) reformulation, which avoids costly conic programming and enables scalable algorithms. In this paper, we develop a simple and efficient alternating direction method of multiplier (ADMM) based solver for the SCLS problem. By introducing a consensus splitting that separates the quadratic objective from the spherical constraint, we obtain an augmented Lagrangian formulation with closed-form updates: the primal quadratic step reduces to solving a fixed shifted linear system, the constraint step is a projection onto the unit sphere, and the dual step is a lightweight scaled ascent. The resulting method has low per-iteration complexity and allows pre-factorization of the constant system matrix for substantial speedups. Experiments demonstrate that the proposed ADMM approach achieves competitive solution quality with significantly improved computational efficiency compared with existing global solvers for SCLS, particularly in sparse and high-dimensional regimes.

Toward Multi-Satellite Cooperative Transmission: A Joint Framework for CSI Acquisition, Feedback, and Phase Synchronization

The stringent link budget, caused by long propagation distances and payload constraints, poses a fundamental bottleneck for single-satellite transmission. Although LEO mega-constellations make multi-satellite cooperative transmission (MSCT), such as distributed precoding (DP), increasingly feasible, its cooperative gains critically rely on stringent time-frequency-phase synchronization (TFP-Sync), which is difficult to maintain under rapid channel variation and feedback latency. To address this issue, this paper proposes a joint CSI acquisition, feedback, and phase-level synchronization (JCAFPS) framework for MSCT. Specifically, to enable reliable, overhead-efficient CSI acquisition, we design a beam-domain adjustable phase-shift tracking reference signal (TRS) transmission scheme, along with criteria for the TRS and CSI-feedback periods. Then, exploiting deterministic orbital motion and dominant LoS propagation, we establish a polynomial model for the temporal evolution of delay and Doppler shift, and derive an OFDM-based multi-satellite signal model under non-ideal synchronization. The analysis reveals that, unlike the single-satellite case, the composite multi-satellite channel exhibits nonlinear time-frequency-varying phase behavior, necessitating symbol- and subcarrier-wise phase precompensation for coherent transmission. Based on these results, we develop a practical closed-loop realization integrating single-TRS-based channel parameter estimation, multi-TRS-based channel prediction, predictive CSI feedback, and user-specific TFP precompensation. Numerical results demonstrate that the proposed framework achieves accurate CSI acquisition and precise TFP-Sync, enabling DP-based dual-satellite cooperative transmission to approach the theoretical 6 dB power gain over single-satellite transmission, while remaining robust under extended prediction durations and enlarged TRS periods.

Deep Learning-Based Multi-Satellite Massive MIMO Transmission: Centralized or Decentralized?

This paper investigates new efficient transmission architectures for multi-satellite massive multiple-input multiple-output (MIMO). We study the weighted sum-rate maximization problem in a multi-satellite system where multiple satellites transmit independent data streams to multi-antenna user terminals, thereby achieving higher throughput. We first adopt a multi-satellite weighted minimum mean square error (WMMSE) formulation under statistical channel state information (CSI), which yields closed-form updates for the precoding and receive vectors. To overcome the high complexity of optimization, we propose a learning-based WMMSE design that integrates tensor equivariance with closed-form recovery, enabling inference with near-optimal performance without iterative updates. Moreover, to reduce inter-satellite signaling overhead incurred by exchanging CSI and precoding vectors in centralized coordination, we develop a decentralized multi-satellite transmission scheme in which each satellite locally infers its precoders rather than receiving from the central satellite. The proposed decentralized scheme leverages periodically available satellite state information, such as orbital positions and satellite attitude, which is inherently accessible in satellite networks, and employs a dual-branch tensor-equivariant network to predict the precoders at each satellite locally. Numerical results demonstrate that the proposed multi-satellite transmission significantly outperforms single-satellite systems in sum rate; the decentralized scheme achieves sum-rate performance close to the centralized schemes while substantially reducing computational complexity and inter-satellite overhead; and the learning-based schemes exhibit strong robustness and scalability across different scenarios.

Digital-Twin-Aided Dynamic Spectrum Sharing and Resource Management in Integrated Satellite-Terrestrial Networks

The explosive growth in wireless service demand has prompted the evolution of integrated satellite-terrestrial networks (ISTNs) to overcome the limitations of traditional terrestrial networks (TNs) in terms of coverage, spectrum efficiency, and deployment cost. Particularly, leveraging LEO satellites and dynamic spectrum sharing (DSS), ISTNs offer promising solutions but face significant challenges due to diverse terrestrial environments, user and satellite mobility, and long propagation LEO-to-ground distance. To address these challenges, digitial-twin (DT) has emerged as a promising technology to offer virtual replicas of real-world systems, facilitating prediction for resource management. In this work, we study a time-window-based DT-aided DSS framework for ISTNs, enabling joint long-term and short-term resource decisions to reduce system congestion. Based on that, two optimization problems are formulated, which aim to optimize resource management using DT information and to refine obtained solutions with actual real-time information, respectively. To efficiently solve these problems, we proposed algorithms using compressed-sensing-based and successive convex approximation techniques. Simulation results using actual traffic data and the London 3D map demonstrate the superiority in terms of congestion minimization of our proposed algorithms compared to benchmarks. Additionally, it shows the adaptation ability and practical feasibility of our proposed solutions.

Fundamental Limits for Near-Field Sensing -- Part II: Wide-Band Systems

Near-field sensing with extremely large-scale antenna arrays (ELAAs) in practical 6G systems is expected to operate over broad bandwidths, where delay, Doppler, and spatial effects become tightly coupled across frequency. The purpose of this and the companion paper (Part I) is to develop the unified Cram'er--Rao bounds (CRBs) for sensing systems spanning from far-field to near-field, and narrow-band to wide-band. This paper (Part II) derives fundamental estimation limits for a wide-band near-field sensing systems employing orthogonal frequency-division multiplexing signaling over a coherent processing interval. We establish an exact near-field wide-band signal model that captures frequency-dependent propagation, spherical-wave geometry, and the intrinsic coupling between target location and motion parameters across subcarriers and slow time. Similar as Part I using the Slepian--Bangs formulation, we derive the wide-band Fisher information matrix and the CRBs for joint estimation of target position, velocity, and radar cross-section, and we show how wide-band information aggregates across orthogonal subcarriers. We further develop tractable far-field and near-field approximations which provide design-level insights into the roles of bandwidth, coherent integration length, and array aperture, and clarify when wide-band effects. Simulation results validate the derived CRBs and its approximations, demonstrating close agreement with the analytical scaling laws across representative ranges, bandwidths, and array configurations.

Multi-Satellite Multi-Stream Beamspace Massive MIMO Transmission

This paper studies multi-satellite multi-stream (MSMS) beamspace transmission, where multiple satellites cooperate to form a distributed multiple-input multiple-output (MIMO) system and jointly deliver multiple data streams to multi-antenna user terminals (UTs), and beamspace transmission combines earth-moving beamforming with beam-domain precoding. For the first time, we formulate the signal model for MSMS beamspace MIMO transmission. Under synchronization errors, multi-antenna UTs enable the distributed MIMO channel to exhibit higher rank, supporting multiple data streams. Beamspace MIMO retains conventional codebook based beamforming while providing the performance gains of precoding. Based on the signal model, we propose statistical channel state information (sCSI)-based optimization of satellite clustering, beam selection, and transmit precoding, using a sum-rate upper-bound approximation. With given satellite clustering and beam selection, we cast precoder design as an equivalent covariance decomposition-based weighted minimum mean square error (CDWMMSE) problem. To obtain tractable algorithms, we develop a closed-form covariance decomposition required by CDWMMSE and derive an iterative MSMS beam-domain precoder under sCSI. Following this, we further propose several heuristic closed-form precoders to avoid iterative cost. For satellite clustering, we enhance a competition-based algorithm by introducing a mechanism to regulate the number of satellites serving certain UT. Furthermore, we design a two-stage low-complexity beam selection algorithm focused on enhancing the effective channel power. Simulations under practical configurations validate the proposed methods across the number of data streams, receive antennas, serving satellites, and active beams, and show that beamspace transmission approaches conventional MIMO performance at lower complexity.

Unlocking Symbol-Level Precoding Efficiency Through Tensor Equivariant Neural Network

Although symbol-level precoding (SLP) based on constructive interference (CI) exploitation offers performance gains, its high complexity remains a bottleneck. This paper addresses this challenge with an end-to-end deep learning (DL) framework with low inference complexity that leverages the structure of the optimal SLP solution in the closed-form and its inherent tensor equivariance (TE), where TE denotes that a permutation of the input induces the corresponding permutation of the output. Building upon the computationally efficient model-based formulations, as well as their known closed-form solutions, we analyze their relationship with linear precoding (LP) and investigate the corresponding optimality condition. We then construct a mapping from the problem formulation to the solution and prove its TE, based on which the designed networks reveal a specific parameter-sharing pattern that delivers low computational complexity and strong generalization. Leveraging these, we propose the backbone of the framework with an attention-based TE module, achieving linear computational complexity. Furthermore, we demonstrate that such a framework is also applicable to imperfect CSI scenarios, where we design a TE-based network to map the CSI, statistics, and symbols to auxiliary variables. Simulation results show that the proposed framework captures substantial performance gains of optimal SLP, while achieving an approximately 80-times speedup over conventional methods and maintaining strong generalization across user numbers and symbol block lengths.

Demonstrator Testbed for Effective Precoding in MEO Multibeam Satellites

The use of communication satellites in medium Earth orbit (MEO) is foreseen to provide quasi-global broadband Internet connectivity in the coming networking ecosystems. Multi-user multiple-input single-output (MU-MISO) digital signal processing techniques, such as precoding, emerge as appealing technological enablers in the forward link of multi-beam satellite systems operating in full frequency reuse (FFR). However, the orbit dynamics of MEO satellites pose additional challenges that must be carefully evaluated and addressed. This work presents the design of an in-lab testbed based on software-defined radio (SDR) platforms and the corresponding adaptations required for efficient precoding in a MEO scenario. The setup incorporates a precise orbit model and the radiation pattern of a custom-designed direct radiating array (DRA). We analyze the main impairments affecting precoding performance, including Doppler shifts and payload phase noise, and propose a synchronization loop to mitigate these effects. Preliminary experimental results validate the feasibility and effectiveness of the proposed solution.

Interference in Spectrum-Sharing Integrated Terrestrial and Satellite Networks: Modeling, Approximation, and Robust Transmit Beamforming

This paper investigates robust transmit (TX) beamforming from the satellite to user terminals (UTs), based on statistical channel state information (CSI). The proposed design specifically targets the mitigation of satellite-to-terrestrial interference in spectrum-sharing integrated terrestrial and satellite networks. By leveraging the distribution information of terrestrial UTs, we first establish an interference model from the satellite to terrestrial systems without shared CSI. Based on this, robust TX beamforming schemes are developed under both the interference threshold and the power budget. Two optimization criteria are considered: satellite weighted sum rate maximization and mean square error minimization. The former achieves a superior achievable rate performance through an iterative optimization framework, whereas the latter enables a low-complexity closed-form solution at the expense of reduced rate, with interference constraints satisfied via a bisection method. To avoid complex integral calculations and the dependence on user distribution information in inter-system interference evaluations, we propose a terrestrial base station position-aided approximation method, and the approximation errors are subsequently analyzed. Numerical simulations validate the effectiveness of our proposed schemes.

Statistical CSI-Based Distributed Precoding Design for OFDM-Cooperative Multi-Satellite Systems

This paper investigates the design of distributed precoding for multi-satellite massive MIMO transmissions. We first conduct a detailed analysis of the transceiver model, in which delay and Doppler precompensation is introduced to ensure coherent transmission. In this analysis, we examine the impact of precompensation errors on the transmission model, emphasize the near-independence of inter-satellite interference, and ultimately derive the received signal model. Based on such signal model, we formulate an approximate expected rate maximization problem that considers both statistical channel state information (sCSI) and compensation errors. Unlike conventional approaches that recast such problems as weighted minimum mean square error (WMMSE) minimization, we demonstrate that this transformation fails to maintain equivalence in the considered scenario. To address this, we introduce an equivalent covariance decomposition-based WMMSE (CDWMMSE) formulation derived based on channel covariance matrix decomposition. Taking advantage of the channel characteristics, we develop a low-complexity decomposition method and propose an optimization algorithm. To further reduce computational complexity, we introduce a model-driven scalable deep learning (DL) approach that leverages the equivariance of the mapping from sCSI to the unknown variables in the optimal closed-form solution, enhancing performance through novel dense Transformer network and scaling-invariant loss function design. Simulation results validate the effectiveness and robustness of the proposed method in some practical scenarios. We also demonstrate that the DL approach can adapt to dynamic settings with varying numbers of users and satellites.