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Holographic Metasurface-Based Beamforming for Multi-Altitude LEO Satellite Networks
Qingchao Li, Mohammed El-Hajjar, Kaijun Cao, Chao Xu, Harald Haas, Lajos Hanzo
TL;DR
Multi-altitude LEO networks need beamforming that handles inter-satellite interference without the overhead of acquiring full CSI, while conventional arrays face scalability and power limitations. The paper combines holographic serving-link beamforming with MMSE digital combining and proposes a stochastic-geometry-based low-complexity alternative. The architecture achieves higher throughput than a same-size state-of-the-art antenna array, while dense deployments benefit from mutual-coupling-aware design.
Problem
Multi-satellite beamforming must address inter-satellite interference and the bandwidth and computational overhead of full CSI acquisition, alongside conventional arrays’ scalability and power limitations.
Method
The paper maximizes serving-satellite channel gain with a holographic beamformer and uses MMSE digital combining, including a stochastic-geometry design based on satellite-distribution statistics.
Results
The proposed hybrid architecture achieves higher throughput than a same-size state-of-the-art antenna array, while distribution-based MMSE matches full-CSI throughput in dense deployments and outperforms MRC.
Takeaways & Limitations
Mutual coupling should be included in beamforming design because dense holographic metasurface placement can reduce throughput when its effects are ignored.
Abstract
from arXiv · showhide
Low Earth Orbit (LEO) satellite networks are capable of improving the global Internet service coverage. In this context, we propose a hybrid beamforming design for holographic metasurface based terrestrial users in multi-altitude LEO satellite networks. Firstly, the holographic beamformer is optimized by maximizing the downlink channel gain from the serving satellite to the terrestrial user. Then, the digital beamformer is designed by conceiving a minimum mean square error (MMSE) based detection algorithm for mitigating the interference arriving from other satellites. To dispense with excessive overhead of full channel state information (CSI) acquisition of all satellites, we propose a low-complexity MMSE beamforming algorithm that only relies on the distribution of the LEO satellite constellation harnessing stochastic geometry, which can achieve comparable throughput to that of the algorithm based on the full CSI in the case of a dense LEO satellite deployment. Furthermore, it outperforms the maximum ratio combining (MRC) algorithm, thanks to its inter-satellite interference mitigation capacity. The simulation results show that our proposed holographic metasurface based hybrid beamforming architecture is capable of outperforming the state-of-the-art antenna array architecture in terms of its throughput, given the same physical size of the transceivers. Moreover, we demonstrate that the beamforming performance attained can be substantially improved by taking into account the mutual coupling effect, imposed by the dense placement of the holographic metasurface elements.
I. INTRODUCTION
The paper motivates holographic metasurface beamforming for multi-altitude LEO networks by addressing inter-satellite interference, full-CSI overhead, and conventional-array scalability limits. It proposes a statistical MMSE hybrid architecture for terrestrial users and evaluates its throughput benefits under dense deployments and mutual coupling.
- B. Motivation: Dense LEO constellations create inter-satellite interference, while full CSI acquisition imposes substantial bandwidth and computational overhead.Existing multi-satellite beamforming designs are described as overlooking interference and requiring full channel knowledge across satellite links.
- B. Motivation: Holographic metasurfaces offer high directional gain with compact hardware, addressing conventional antenna arrays’ scalability, cost, and power-consumption limitations.The proposed architecture places a holographic metasurface at the terrestrial user for the multi-altitude LEO downlink.
- B. Motivation: The hybrid design separates non-convex optimization into serving-link holographic beamforming and MMSE digital combining for interference mitigation.The holographic beamformer maximizes serving-satellite channel gain, while the digital receiver combines signals under the MMSE criterion.
- B. Motivation: The low-complexity MMSE method uses the average number and spatial distribution of visible interfering satellites rather than full CSI.Stochastic geometry supplies the constellation statistics used to reduce channel-estimation complexity.
- B. Motivation: The reported results show higher throughput than a state-of-the-art antenna array, MMSE gains over MRC, comparable dense-deployment throughput to full CSI, and benefits from modeling mutual coupling.Mutual-coupling-aware beamforming compensates for throughput loss caused by dense element placement; ignoring coupling can degrade performance as element spacing decreases.
- II. SYSTEM MODEL: The system models satellites at different altitudes within a spherical shell, with users served by the nearest satellite and all co-resource satellites acting as interferers.Satellites are uniformly distributed in a three-dimensional binomial point process, and the nearest satellite defines service.
A. Holographic Metasurface-Based Beamforming
The architecture models satellite-to-user channels through holographic metasurface elements, including path loss, shadowed-Rician fading, and mutual coupling. The metasurface structure uses software-controlled elements connected through microstrips and feeds.
- Holographic metasurface structure: Each microstrip contains a feed, waveguide, and N software-controlled subwavelength metamaterial elements.The waveguide carries electromagnetic propagation, while the elements adjust electromagnetic-wave coefficients through a controller.
- Element coefficients: The coefficient β_n^(m) represents the weighting applied by the nth reconfigurable element on the mth microstrip.The elements follow a Lorentzian-constrained phase model, with phase parameter n ∈ [0, 2π).
- Channel model: The satellite-to-microstrip channel f^(l,m) combines link attenuation and small-scale fading for each satellite–user connection.The link attenuation depends on wavelength, rain attenuation, antenna gain, path-loss exponent, and satellite–user distance; fading follows a shadowed-Rician model.
- Mutual coupling: Mutual coupling is modeled with a Z-parameter-based coupling matrix C, using fixed 50-Ohm antenna and load impedances.The mutual impedance matrix entries depend on inter-element distances and are characterized using cosine and sine integrals.
C. Distribution of Satellite Constellation
This section derives geometric distributions for visible satellites in a multi-altitude constellation. It characterizes satellite distances, serving-satellite visibility, and interfering-satellite distances using visible-shell volumes and conditional distributions.
- Visible-shell geometry: The visible region Ω′ is modeled as a spherical cap or shell within the satellite constellation geometry.The analysis begins by deriving the probability that a specific satellite lies in the visible region.
- Satellite-distance distribution: Theorem 1 gives the CDF of the distance from a specific visible-shell satellite to the terrestrial user under two altitude-dependent cases.The cases are separated by the condition H1(2Re + H1) ≤ H2 and its complement.
- Satellite-distance distribution: Corollary 1 obtains the corresponding PDF by differentiating the satellite-distance CDF.The PDF is specified over piecewise distance intervals determined by the shell geometry.
- Serving satellite: The serving satellite is defined as the satellite with minimum distance to the terrestrial user, and its visible-region probability is derived from the shell probability.The derivation uses the visible-shell volume V′ and the total shell volume V.
- Interfering satellites: Conditioned on serving-satellite distance D0 = d0, the analysis derives the probability and conditional PDF for other visible satellites that act as interferers.Corollary 3 gives the probability that another satellite lies within the visible shell, while Theorem 2 gives its conditional distance distribution.
III. HYBRID HOLOGRAPHIC AND DIGITAL BEAMFORMING DESIGN
The hybrid design jointly optimizes the holographic metasurface beamformer and digital combiner to maximize throughput while mitigating interference from other satellites. The non-convex problem is decomposed into channel-gain maximization and MMSE-based digital detection.
- System design: The receiver uses a holographic beamformer and a digital beamformer to maximize the throughput of the considered LEO system.The digital combiner v operates on the RF-chain outputs.
- Equivalent channel: The baseband channel from each satellite to the RF chains is formed through the metasurface structure and mutual-coupling-aware element responses.The channel formulation uses the block structure of A and the element-level links f′^(l,m).
- Received signal: The received signal contains the serving-satellite signal, inter-satellite interference at the RF chains, and equivalent additive noise.The equivalent RF-chain noise is w′ = ACw with covariance determined by the coupling matrix.
- Optimization problem: The design optimizes the digital beamformer v and metasurface coefficient matrix B subject to the throughput objective.The resulting optimization problem is non-convex.
- Alternating design: The non-convex problem is decoupled into holographic channel-gain maximization and MMSE digital-beamformer optimization.The first subproblem strengthens the serving link, while the second reduces inter-satellite interference.
A. Holographic Beamformer
The holographic beamformer is designed by maximizing the baseband channel gain between the serving satellite and the terrestrial user. The resulting problem is reformulated into subproblems with a closed-form solution for the final stage.
- Beamformer objective: Maximizing the baseband channel gain improves directional beam shaping and strengthens the serving satellite’s signal against satellite-to-ground path loss.The channel-gain objective is used to formulate the holographic-beamformer optimization problem.
- Phase constraints: The metasurface phase variables are constrained to n ∈ [0, 2π) for each indexed element.The element indices cover n = 1, 2, · · ·, N.
- Problem reformulation: The channel-gain objective ∥h(0)∥2 is reformulated to enable tractable optimization of the holographic beamformer.The reformulation leads from problem P2 to a further subproblem formulation.
- Closed-form solution: The final subproblem P3 has a closed-form solution for the holographic-beamformer design.The solution is stated after fixing the microstrip index m over its allowed range.
B. Digital Beamformer
The digital beamformer design follows holographic beamformer optimization by forming baseband channels and applying MMSE combining to mitigate interference from other satellites.
- The holographic beamformer first determines the baseband channel between the serving satellite and terrestrial user.
- Interfering-satellite links are characterized separately to represent inter-satellite interference at the terrestrial user.
- The digital receiver combining vector is then designed using MMSE methods with different channel-state-information requirements.
1) MMSE RC method based on full CSI:
The full-CSI MMSE receiver assumes that the terrestrial user knows the channels from the serving and all interfering satellites, then designs the combining vector accordingly.
- Full-CSI MMSE combining assumes that the terrestrial user acquires channel state information from the serving and every interfering satellite.
- The MMSE receiver-combining vector is designed using the complete channel information from the satellite links.
- The resulting combining design determines the attainable throughput of the full-CSI method.
2) MMSE RC method based on the distribution of satellites:
The statistical MMSE receiver replaces individual interfering-satellite CSI with constellation-level distributional information derived using stochastic geometry.
- The proposed statistical MMSE method uses only the average number and spatial distribution of interfering satellites in the visible region.
- It avoids requiring individual interfering-satellite channels and precise azimuth and elevation positions, reducing channel-estimation complexity.
- The method represents interference through the covariance matrix of baseband channels from visible interfering satellites to the terrestrial user.
- The average small-scale fading term is defined for interfering links conditioned on the serving-satellite distance, with separate expressions for path-loss exponents.
- Given the satellite distribution, the hybrid beamforming throughput is subsequently formulated from the statistical receiver design.
C. Computational Complexity of the RC Methods
The statistical MMSE receiver reduces computational and CSI-acquisition demands relative to full-CSI MMSE, supporting its use in dense LEO deployments.
- The complexity analysis counts floating-point multiplications and divisions for combining-vector calculation and information recovery.
- For full CSI, forming the serving-channel outer product and covariance-related terms contributes an average [(|A| −1)PI + 1]M 2 multiplications.
- Both methods require M floating-point multiplications for information recovery, while matrix inversion adds M 2 multiplications and M divisions under LDLH decomposition.
- Statistical MMSE requires M 2 multiplications for h(0)h(0)H, while its covariance calculation uses constellation statistics that remain unchanged within each statistical block.
- Full-CSI MMSE requires complete serving- and interfering-satellite CSI and calculation of the full covariance matrix.
- Statistical MMSE computes its combining vector from constellation statistics, avoiding full-CSI acquisition and reducing computational demand.
IV. SIMULATION RESULTS
The simulations evaluate holographic metasurface beamforming across element configuration, mutual coupling, satellite density, path loss, and orbit altitude. Results show that interference-aware beamforming and explicit mutual-coupling modeling are important for throughput.
- Beamforming and mutual coupling: Considering mutual coupling in the beamforming design significantly improves throughput, while the distribution-based MMSE RC method nearly matches full-CSI throughput with lower overhead and complexity.The MMSE RC method also exceeds MRC throughput by mitigating inter-satellite interference.
- Beamforming and mutual coupling: Increasing holographic metasurface elements improves throughput when coupling is considered, but ignoring coupling can degrade throughput as reduced spacing intensifies mutual coupling.More RF chains can further increase throughput at higher hardware and energy cost.
- Physical size and spacing: Decreasing holographic metasurface element spacing improves throughput within a fixed microstrip size, enabling the holographic metasurface to outperform the state-of-the-art antenna array.The comparison uses full-digital beamforming for the antenna array with antenna spacing of λ/2.
- Constellation conditions: When satellites are sparse, increasing their number improves throughput through higher serving-satellite visibility, whereas further increases degrade throughput because of inter-satellite interference.The sparse regime is identified as |A| < 30.
- Constellation conditions: Throughput improves with increasing path loss exponent because the higher exponent mitigates inter-satellite interference.Fig. 8 reports this relationship through the SIR γ0 across beamforming methods.
- Constellation conditions: Altitude effects depend on satellite count: throughput rises with altitude for 10 satellites, is nonmonotonic for 30, and decreases consistently for 300.At high density, increasing altitude exacerbates inter-satellite interference.
V. CONCLUSIONS
The paper proposes hybrid holographic metasurface beamforming for multi-altitude LEO networks, combining serving-link channel-gain optimization with interference-mitigating digital beamforming. Its statistical MMSE RC scheme reduces CSI overhead while retaining comparable dense-deployment throughput, and the architecture benefits from explicit mutual-coupling modeling.
- Contributions: The proposed hybrid design optimizes the holographic beamformer for serving-satellite channel gain and the digital beamformer for interference mitigation.The design targets multi-altitude LEO satellite networks.
- Contributions: The stochastic-geometry MMSE RC scheme achieves comparable throughput to full-CSI processing in dense LEO deployments while reducing CSI acquisition overhead and computational complexity.The scheme relies on statistical information about the LEO satellite constellation.
- Contributions: The holographic metasurface hybrid beamformer achieves higher throughput than the full-digital state-of-the-art antenna-array beamformer for equal transceiver physical size.The conclusion identifies dense element placement as a reason to explicitly include mutual coupling in beamforming design.
APPENDIX B PROOF OF Theorem 2
Appendix B derives conditional distributions and expectations used in the stochastic-geometry analysis of interfering satellites and received channel quantities.
- Conditional distance distribution: The proof derives the conditional distance density of interfering satellites given the serving-satellite distance.The resulting expression is obtained by substituting earlier distance distributions into the conditional density relation.
- Channel and interference statistics: The proof expresses the channel expectation E[h(l)h(l)^H] and the path-loss quantity L(d0)=E[d^-α] using the derived distance distributions.These quantities support the analytical characterization of the interference and received signal.
- Channel and interference statistics: The appendix derives the average number of interfering satellites in the visible region from the preceding conditional distributions.This quantity is denoted as |A′|.