Hexagonal planar arrays are widely utilized in radar and satellite communications due to their efficient aperture tiling and six-fold rotational symmetry. While the technique of Goto provides a theoretical framework to map 1D linear tapers onto hexagonal grids, its traditional implementation relies on closed-form binomial expansions that suffer from numerical instability for large-scale arrays. This letter proposes a robust alternative using Recursive Spatial Mapping. By treating the hexagonal mapping kernel as a discrete six-point spatial operator, the aperture excitations are synthesized via successive 2D convolutions. This approach improves efficiency eliminating factorial-based computations, ensures numerical stability for high-order arrays, and naturally preserves the triangular lattice geometry.

The integration of non-terrestrial networks (NTNs) into future sixth-generation (6G) architectures is essential for achieving truly global, seamless, and resilient connectivity. This article provides a critical perspective on the evolution of massive multiple input multiple output (M-MIMO) and advanced radio resource management (RRM) for broadband satellite systems, focusing on pragmatic, scalable solutions that bridge the gap between theoretical performance and real-world satellite networks deployment. We argue that the traditional terrestrial approach to M-MIMO—relying on instantaneous channel state information and complex adaptive interference mitigation—is fundamentally incompatible with the physics of satellite propagation and high round-trip delays. Instead, following our earlier pragmatic approach, we describe a shift toward beam-switching beamforming over a dense fixed-lattice and heuristic, low-complexity resource allocation. By synthesizing recent advances in transform-domain processing and load-aware regularization, we show how these pragmatic architectures can effectively manage extreme spatial and temporal traffic non-uniformity. Our framework evaluates performance through a user-centric traffic satisfaction metric, ensuring that 6G NTNs can provide consistent quality of service to diverse users, from aeronautical corridors to remote rural areas. Finally, we outline the critical role of flexible payload design and software-defined architectures in enabling the unification of terrestrial and non-terrestrial components, providing a roadmap for a sustainable and economically viable 6G global infrastructure.

The Butler matrix is a foundational beamforming network for generating multiple orthogonal beams from uniform linear arrays. While its topological analogy with the Fast Fourier Transform (FFT) is well known, the physical meaning of its multistage decomposition has not been fully clarified. This paper establishes a unified framework in which the mixed-radix decomposition of a centered Butler matrix and the classical pattern multiplication principle are shown to be two manifestations of the same underlying structure. For arrays with N = PQ elements, the Butler matrix factorization is shown to correspond exactly to the factorization of the array factor into a fine contribution from sparsely populated sub-arrays and a coarse contribution from their phase centers. Within this framework, the twiddle-factor phase-shift layer emerges as a physical alias-suppression mechanism that aligns the nulls of the coarse factor with the grating-lobe replicas of the fine factor. The analysis also identifies parity-dependent solvability conditions and introduces a generalized phase-correction scheme that resolves previously overlooked pointing ambiguities. The resulting interpretation provides an intuitive, analytically rigorous, and scalable methodology for multibeam array and Butler matrix design. This perspective unifies array signal processing and network synthesis, offering an intuitive, analytically tractable, and scalable methodology for multibeam architecture design.

Line-of-Sight (LoS) Massive-MIMO performance is fundamentally governed by the algebraic properties of the channel Gram matrix, which determines spatial multiplexing efficiency and sum-rate capacity. Traditionally analyzed via linear algebra, matrix behavior is often decoupled from antenna geometry. This letter introduces the k-space co-array, a geometric framework bridging this gap by extending the classical spatial co-array into the wave-vector domain. We establish a Dual Heredity principle, revealing that the Gram matrix is a discrete sampling of a virtual array factor that inherits the exact microscopic lattice periodicity and the macroscopic rim of the physical aperture. Mapping the Gram matrix entries directly to these geometric structures reveals how spatial aliasing and rank collapse are governed by difference spatial frequencies in k-space. This enables proactive aperture engineering and low-complexity user scheduling, offering a deterministic path to optimizing spatial degrees of freedom in next-generation LoS communications.

The synthesis of lossless multiport networks is a classic topic of circuit theory, with broad application in various signal distribution and processing systems. This paper presents a method for synthesizing reciprocal lossless N-port networks with an assigned scattering matrix, and reciprocal lossless matched and decoupled 2N-port networks with an assigned transmission matrix, as minimum depth chessboard networks. The proposed approach leverages bilateral factorization techniques, extending existing reductive methods by incorporating a new degree of freedom in the selection process. The resulting chessboard network configuration minimizes network depth and component count, offering improved uniformity in input-output paths. Theoretical findings are validated through numerical implementations, demonstrating high accuracy and stability. This method provides a compact and efficient solution for designing reciprocal lossless multiport networks, with significant potential for applications in diverse fields requiring robust and efficient network architectures.

The use of Hamming window for tapering the analog beamformed subarrays (SAs) is investigated in the context of overlapped hybrid beamformed direct radiating array for broadband satellite communication. Whilst this class of antennas can suppress some of the grating lobes (GLs) associated with conventional nonoverlapped arrays, there are still concerns on interference from remaining GLs. We discuss how tapering the analog SA can complement the GL mitigation strategy and demonstrate that a Hamming window offers a simple closed-form implementation leading to significant reduction of interfering signals. Moreover, by virtue of analytical and numerical results, we derive the configuration of the Hamming window tapering that optimizes the tradeoff between sidelobe suppression and gain losses in the main lobe.

This paper introduces Perfect-PAT codes, a family of polyphase sequences that systematically achieve both perfect periodic autocorrelation and superior aperiodic autocorrelation performance. By exploiting a unique spectral property of the original PAT codes, we propose a generalized phase modulation scheme that renders the sequences perfect (i.e. Constant Amplitude Zero periodic Auto-Correlation- CAZAC) while preserving their excellent aperiodic autocorrelation characteristics. Closed form expressions for the unwrapped phases of the resulting codes are derived, providing a simple, analytical, and systematic method for generating Perfect-PAT sequences. Numerical examples demonstrate that Perfect-PAT codes achieve ideal periodic autocorrelation and maintain superior Merit Factor (MF) and Peak-to-Side-Peak Ratio (PSPR) compared to classical piecewise linear polyphase sequences, including Frank, P1, P2, and Px codes.

Resource distribution in radio networks aims at maximizing spectrum utilization while minimizing interference. In this paper, we consider the problem of uniform radio resource distribution on a periodic grid. We formulate the problem as finding the sublattice configuration that maximises the distance between adjacent resources, crucial for reducing interference and improving throughput performance. Leveraging concepts from lattice theory and discrete geometry, we present an enumerative, parallelizable algorithm to explore all possible sublattices and efficiently identify the optimal configurations. Additionally, we investigate the existence and properties of scaled-rotated sublattices, exploring how different lattice geometries impact optimal solutions. Numerical results demonstrate the effectiveness of the proposed algorithm and highlight insights into optimal sublattice design for various lattice structures. Furthermore, the results are applied to the identification of the beam layout in a fixed multibeam geostationary satellite. Numerical results show that the spectral efficiency of the optimised sublattice is higher than all other sublattices. This work thus advances the field of radio resource distribution and offers practical implications for improving satellite network performance.

This paper demonstrates that the electromagnetic design of rotationally symmetric reactive combiners for solid-state power amplifiers (SSPAs) with enhanced graceful degradation is governed by three key parameters: output-port return loss, inputport return loss, and inter-port coupling. While output-port return loss determines nominal combining efficiency, graceful degradation under single-device failure is governed by two factors: input-port return loss and symmetry of inter-port power redistribution. Under single-device failure conditions, efficiency degradation scales inversely with input-port return loss, asymptotically approaching the behavior of ideal beam-forming network (BFN) combiners. Failure-induced residual power is redistributed among the remaining input ports, increasing RF stress on the surviving MMICs and necessitating isolators with adequate power-handling capability. Optimal fault tolerance is achieved when power injected into any one port is uniformly redistributed among all other ports.

This paper provides an overview of some of the recent technology developments and R&D activities supported by the European Space Agency for next generation satellite communication payloads and earth observation spaceborne instruments based on active antennas and on-board digital processing functions. In particular, for satellite communication payloads, recent technology developments on active antennas for LEO, MEO and GEO missions, including HPA MMICs, are presented. For earth observation instruments, recent active antenna technology developments for synthetic aperture radars and synthetic aperture radiometers are presented. An introduction to R&D activities in the area of digital signal processing and digital beamforming, is also provided.

Array-fed reflectors are widely used for multibeam satellite systems, but their efficiency is often limited by the highly nonuniform power distribution across the feed array. This, in turn, calls for increased back-off in the operation of the high power amplifiers and therefore penalising the payload efficiency. In this work, we propose the use of shaped reflectors as means of mitigating this limitation. We demonstrate that by introducing controlled surface deformations in the paraboloidal surface, it is possible to improve the uniformity of the power distribution across the feed array in a multibeam coverage scenario. The problem is formulated as a multi-objective optimisation balancing average directivity and feed power uniformity. We then propose a dedicated four-stage workflow to efficiently solve this problem: three fast stages using a Fast Fourier Transform (FFT)-based geometrical optics solver, followed by a final refinement with a physical optics solver. Two basis functions, Zernike polynomials and cubic B-splines, are proposed to describe the surface profile, with the former achieving more uniform power distribution using fewer terms. Results demonstrate up to 2.1 dB reduction in high-power amplifier peak-to-peak dynamics with negligible loss in beam gain, providing a practical path toward more efficient multibeam satellite payloads.

This paper addresses optimal frequency reuse in multibeam satellite systems featuring regular beam lattices, such as those employed in GEO, MEO, and LEO constellations. To maximize spectral efficiency while mitigating co-channel interference, the study formulates the resource assignment problem as an optimization over lattice substructures. We apply the maximum same-colour nearest-neighbour sublattice (MASCONDS) algorithm, a globally optimal method that identifies the sublattice partition maximizing the minimum co-channel distance for any lattice and reuse factor. Numerical analyses with a hexagonal lattice for a non-canonical reuse factor validate the approach using a GEO communication payload with a direct radiating array model operating in Ka-band. The MASCONDS solution consistently yields the highest carrier-to-interference ratio across all configurations. These results demonstrate the method's effectiveness for advanced satellite payloads constrained by digital processor port counts.

This paper describes the structure of a reconfigurable digital beamforming network, which offers significant improvements in implementation complexity and cost, while maintaining a high degree of flexibility. This architecture supports a multi-beam array antenna and allows complete flexibility in steering each individual beam produced by this antenna, achieving computational complexity comparable to that of the Fast Fourier Transform as the number of beams and antenna elements change. The proposed architecture is built upon the theory behind the Non-Uniform Fast Fourier Transform and is demonstrated for linear array geometries with regular or non-regular element positions. Fixed-point simulations, with a view to real-time hardware implementation, demonstrate effective reduction in the computational complexity of the reconfigurable digital beamforming network and its full flexibility.

This work presents a fast and general formulation for the directivity computation of planar antenna arrays composed of radiating elements modeled as cosqθ transmitters. The main novelty in the proposed formulation as compared to the literature is the fact that an arbitrary number of different radiating elements, each one characterized by a different q-factor, can be considered. The proposed framework enables the evaluation and validation of array architectures previously introduced in the literature, including concentric-ring configurations with tapered element sizes. A unified expression is derived in which the directivity depends solely on two inputs: the array factor and the class-dependent radiating element size. This reduced representation allows real-time computation of directivity, even for electrically large planar arrays comprising multiple rings and heterogeneous element classes. The resulting computational complexity is significantly lower than that of classical integral-based approaches traditionally used for directivity evaluation. Numerical results confirm that the proposed formulation yields accurate directivity values that are fully consistent with existing analytical and numerical results reported in the literature.

The evolution of satellite communication systems is increasingly driven by the need for high-performance, flexible, and efficient solutions, particularly in the design and implementation of circuits and systems for next-generation satellite payloads. Software defined payloads leveraging on On-Board Digital Signal Processing (OBP), Multi-Beam Antennas (MBAs) and Beam-Forming Networks (BFNs) are pivotal technologies in addressing the challenges posed by emerging communication standards, such as 5G/6G, as well as advanced satellite communication, navigation, and remote sensing systems. The realization of these flexible payloads presents significant challenges for circuits and systems designers. These include developing high-frequency, RF and mixed-signal circuits for active phased arrays, designing power-efficient digital signal processing architectures for real-time routing and beamforming, ensuring robust power management systems, and achieving extreme miniaturization through advanced VLSI and SoC integration while maintaining radiation hardness and efficient thermal management. These demands, however, create vast opportunities for innovation in novel analog and digital circuit architectures, efficient data converters, high-speed interfaces, and integrated system-on-chip solutions that leverage advanced semiconductor processes for space-grade reliability and performance. This synergy between evolving market demands, cutting-edge payload technologies, and the continuous innovation in circuit and system design is transforming SATCOM systems, and is essential for industry’s continued growth and relevance.

Overlapping strategies for hybrid beamformed direct radiating arrays have been studied in the past decades as a means to improve the gain of a directive beam. It is widely known that overlapping subarrays guarantees the suppression or the partial mitigation of some unwanted grating lobes. However, such techniques come with a cost, either in terms of hardware or in terms of digital complexity. Performance-wise, the overlapping strategies have also shown particular cases where the scanning improvements were not evident. This paper presents a study of the general overlapping principles in hybrid analogue-digital beamformed direct radiating arrays for the case of a GEO satellite application. This study allows to identify cases where the overlapping technique is beneficial compared to a non-overlapped array.

Direct radiating arrays (DRAs) present favorable solutions for high-throughput flexible coverage in geostationary (GEO) broadband satellite missions. The ultimate constraint in these architectures is the high number of digitally controlled antenna ports, which renders fully digital architectures impractical for the immediate future. Instead, hybrid analog–digital DRAs are being considered as a promising trade-off in terms of performance/flexibility and digital processing demands. These architectures commonly involve subarrays with analog beamforming, which form broad (regional) beams, which are then digitally beamformed at a second level to produce a multitude of narrow beams used for broadband connectivity. Due to the large size of the subarrays, these architectures are subject to undesired grating lobes that can lead to interference and reduce the gain of the main beam, thereby compromising overall performance. Partial mitigation of the grating lobes is attainable by subarray overlapping. This paper presents a comparative assessment of three different hybrid analog–digital DRA architectures in terms of the coverage characteristics and discusses their practical implementation. It is demonstrated that improved performance can be achieved by subarray overlapping with some additional analog hardware complexity but otherwise maintaining the number of digitally controlled antenna ports.

The paper describes active array antennas and beamforming networks (BFN) exploiting a two-level analogue-digital beamforming scheme to reduce the complexity of conventional active arrays based on a fully analogue or a fully digital beamforming. Fully digital BFNs offer clear advantages for meeting the challenging reconfigurability requirements of satellite operators, however the key drawback of these solutions is related to the complexity of the digital BFN implementation, requiring often unaffordable mass and power consumption/dissipation resources. In order to combine the advantages of both analogue and digital beamforming schemes, hybrid active antennas have been introduced. Direct radiating arrays with hybrid beamforming adopt two or more levels of beamforming: one being analogue and the other one digital. While active antennas based on hybrid beamforming schemes have been significantly used for ground systems, only recently they have been considered for satellite systems.

Modern mobile satellite networks rely on advanced onboard antenna systems, used in both large geostationary earth orbit (GEO) satellites and smaller low earth orbit (LEO) satellites in mega-constellations. Current research is exploring coordinated formations or swarms of small satellites with simple antennas to create a large distributed mega-antenna in space. These systems enable advanced network configurations using massive multiple-input multiple-output (M-MIMO) technologies, with the ultimate goal of extending the cell-free M-MIMO (CF-MIMO) approach, proposed for terrestrial networks in beyond-5G and 6G, to wide-area satellite networks. This article focuses on satellite CF-MIMO, considering two antenna configurations: 1) formation of arrays (FoA), with a large closely spaced antenna at S-band in GEO (single FoA with regional or continent-wide coverage, with hundreds of satellites with spacing up to 10 km) or LEO (constellation of few closely spaced satellites); and 2) (mega)constellation of LEO satellites in different orbital planes, with spacing up to 100 km. In such scenarios, we assess the applicability and the relative performance of CF-MIMO technology to a wide-area satellite network through extensive simulation, also including critical system issues such as array geometry, power normalization, synchronization, and radio resource management (RRM). We show that system throughput is maximized by minimizing satellite spacing and employing regular FoA geometries. For both GEO and LEO cases, the potential gains offered by minimum-mean-square-error precoding are contingent upon tight satellite synchronization and minimal intersatellite spacing. Furthermore, adopting an effective RRM algorithm is crucial for achieving optimal performance under both uniform and nonuniform traffic distributions.

The location of conjugate points plays a crucial role in understanding and predicting certain radiation characteristics of reflector antenna systems. In this manuscript, we present an analytical approach for determining the location of conjugate points for both broadside radiation and beam scanning. In the 2D case, our methodology is based on the well-known mirror formula. Moving to the 3D case, we demonstrate that for broadside radiation, reflected rays converge to a single point. However, during beam scanning, incident rays from different planes converge to different points after reflection. These points can be analytically determined by locally approximating the reflector as either an ellipsoid or a hyperboloid. Our findings highlight the limitations of the conjugate point model for larger scan angles.