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

• LEO integration offers low latency, wide coverage, and spatial diversity but introduces dynamic topology and intensive spectrum reuse.Source: §I · p. 1
• RA steering reshapes effective gains, spatial separability, and interference exposure, while satellite selection determines the received multi-satellite channel composition.Source: §I · p. 2
• The proposed two-timescale formulation jointly optimizes epoch-held serving sets and RA boresight trajectories under time-varying geometry, external interference, and mechanical reconfiguration constraints.Source: §I · p. 2, A. Two-Timescale Operation Protocol · p. 3
• With fixed RA trajectories, the satellite-selection objective is monotone submodular because marginal capacity depends on effective strength and spatial complementarity.Source: §I · p. 2
• Discrete MM selection uses an incumbent-tight modular surrogate, while slew-feasible Riemannian RA updates are organized in a parallel alternating framework with nondecreasing throughput and objective convergence.Source: §I · p. 2
• Simulations identify an optimal strength–separability balance: selection matters most when underloaded or actuator-limited, whereas RA shaping matters more near full spatial loading.Source: §I · p. 2

• Each RA element has an epoch-dependent boresight in the GS-centered ENU frame, with feasible orientations restricted to a spherical cap around the array normal.Source: 1) RA Configuration and Directional Gain · p. 4
• The maximum steering angle from the nominal array normal limits the feasible RA steering region.Source: 1) RA Configuration and Directional Gain · p. 4
• Finite actuation speed limits angular displacement between consecutive epoch boresights during each guard interval.Source: 1) RA Configuration and Directional Gain · p. 4
• Epoch-1 boresights are pre-positioned before observation and therefore avoid an initial slew constraint.Source: 1) RA Configuration and Directional Gain · p. 4
• The front-side cosine-power pattern models direction-dependent receive gain; increasing the directivity exponent narrows the main lobe and raises peak gain.Source: 1) RA Configuration and Directional Gain · p. 4

• Satellite-to-GS channels combine RA response, transmit directivity, large-scale propagation, and array-induced phase differences.Source: 2) Satellite-to-GS Channel Model · p. 4
• Time-varying satellite geometry controls propagation gain and array phase, while RA boresights provide element-wise control of directional gains.Source: 2) Satellite-to-GS Channel Model · p. 4
• External interferers are independently operated, uncontrolled satellites whose effective channels depend on geometry, link parameters, and RA-dependent receive gains.Source: D. Multi-Satellite Reception and Sum… · pp. 4–5
• The GS selects at most Kmax serving satellites per epoch and uses one RF chain per RA element with fully digital reception.Source: D. Multi-Satellite Reception and Sum… · p. 4
• MMSE-SIC computes slot sum capacity from instantaneous CSI, while satellite selection and RA control use epoch-level predicted geometry and large-scale channel and interference information.Source: D. Multi-Satellite Reception and Sum… · p. 5
• Scheduling more than M satellites can still increase sum capacity, but streams beyond M add no spatial degrees of freedom and increase acquisition and decoding overhead.Source: D. Multi-Satellite Reception and Sum… · p. 5

E. Problem Formulation

• The effective-throughput problem jointly optimizes epoch-level serving sets and the complete RA trajectory over time-varying intra-epoch geometry.Serving sets and boresights remain fixed within epochs, while effective channels vary across constituent slots.Source: E. Problem Formulation · p. 5
• The formulation is mixed discrete–continuous and nonconvex because satellite selection, RA-dependent channel gains, and inter-epoch slew constraints are coupled.Boresights must remain within feasible steering regions, and consecutive configurations are limited by available reconfiguration time.Source: E. Problem Formulation · p. 5
• Transmit powers are fixed at their maximum values because jointly decoded sum capacity is monotonically nondecreasing in each satellite’s transmit power.The resulting focus is satellite selection and RA control rather than power optimization.Source: E. Problem Formulation · p. 5

A. Geometric Insights for Multi-Satellite Reception

• Interference whitening converts interference-plus-noise covariance to identity, making whitened channel norms and mutual alignment interference-aware strength and separability measures.The representation is equivalent because the whitening transformation is invertible.Source: A. Geometric Insights for… · pp. 5–6
• For two satellites, capacity contains individual strengths α1 and α2 plus a complementarity term α1α2(1 −¯ρ^2) that decreases with channel coherence.Fully aligned channels lose the complementarity contribution, whereas orthogonal channels provide two separable spatial modes.Source: A. Geometric Insights for… · p. 6
• At low SNR, individual channel strengths dominate first-order capacity, while at high SNR spatial separability becomes inseparable from channel strength.Thus, individually strong satellites may still yield limited sum capacity when their whitened channels are highly aligned.Source: A. Geometric Insights for… · p. 6
• The paper therefore jointly designs satellite selection and RA control to balance effective channel strength, spatial orientations, and interference susceptibility.Selection determines the channel set, while RA steering modifies directional gains and relative channel orientations.Source: §III · p. 5, A. Geometric Insights for… · p. 6
• Clustered arrival directions facilitate simultaneous directional-gain alignment but reduce separability, whereas widely separated directions improve separability but hinder simultaneous high-gain alignment.RA steering can partially alleviate this tradeoff by reshaping directional gains.Source: A. Geometric Insights for… · p. 6

B. Optimization for Satellite Selection

• With the RA trajectory fixed, effective throughput decomposes into independent epoch-level selection objectives.Each epoch objective is defined over candidate serving sets for that epoch.Source: B. Optimization for Satellite Selection · p. 6
• Because both the objective and constraints are separable across epochs, all epoch-level satellite-selection problems can be solved independently and in parallel.Each resulting problem is a cardinality-constrained set optimization.Source: B. Optimization for Satellite Selection · p. 6
• The epoch-level selection problem is generally NP-hard, so the method derives conditional marginal gains and establishes monotone submodularity before constructing a modular surrogate.The surrogate is optimized within a discrete Minorization-Maximization framework.Source: B. Optimization for Satellite Selection · p. 6

1) Marginal-Gain Characterization and Submodularity:

• Adding a candidate satellite contributes a rank-one term to the current received covariance, defining its conditional marginal throughput gain.The marginal gain is the objective difference between the enlarged and current serving sets.Source: 1) Marginal-Gain Characterization and… · p. 6
• The marginal-gain expression separates candidate interference-whitened strength from overlap with the selected-channel subspace and its associated separability loss.Large gains favor satellites that are both strong and spatially complementary to the current set.Source: 1) Marginal-Gain Characterization and… · pp. 6–7
• The epoch-level throughput function Fℓ(S) is monotone and submodular on the candidate satellite set.Monotonicity gives nonnegative marginal gains, while submodularity means marginal contributions diminish as the serving set grows.Source: 1) Marginal-Gain Characterization and… · p. 7
• The proof establishes diminishing returns by comparing received covariance matrices for nested serving sets and applying positive-definite order reversal under inversion.This relation is substituted into the marginal-gain expression to prove submodularity.Source: 1) Marginal-Gain Characterization and… · p. 7
• Monotonicity implies selecting all candidates when their number is within the limit, or a set of cardinality Kmax otherwise.Submodularity additionally makes each candidate’s value dependent on the satellites already selected.Source: 1) Marginal-Gain Characterization and… · p. 7

• The incumbent elements are ordered greedily by conditional marginal gains relative to the current chain prefix.Source: 2) Modular Surrogate Construction and… · p. 7
• Each chain weight is an exact conditional marginal gain evaluated along the permutation and lower-bounds the corresponding gain for any subset of preceding elements.Source: 2) Modular Surrogate Construction and… · p. 7
• The resulting modular surrogate is a lower bound on the selection objective and is tight at the incumbent set.Source: 2) Modular Surrogate Construction and… · p. 7
• Maximizing the surrogate under the cardinality constraint selects the Kmax candidates with the largest chain weights.Source: 2) Modular Surrogate Construction and… · p. 7
• The discrete MM algorithm produces a nondecreasing epoch-throughput sequence whose objective values are guaranteed to converge.Source: 2) Modular Surrogate Construction and… · pp. 7–8

• The RA update reshapes effective channel strengths and spatial relationships after satellite serving sets are fixed.Source: C. Optimization for RA Trajectory · p. 8
• Although the objective separates across epochs, inter-epoch slew constraints create chain-structured coupling.Source: C. Optimization for RA Trajectory · p. 8
• The coupled updates are organized into a two-color parallel scheme with neighboring epochs held fixed.Source: C. Optimization for RA Trajectory · p. 8

• The channel coefficient of each source is first expressed in terms of the RA element orientation.Source: 1) Epoch-Level RA Gradient · p. 8
• The Euclidean gradient of effective throughput is derived using d ln det X = tr(X^-1dX).Source: 1) Epoch-Level RA Gradient · p. 8
• The gradient accounts for scheduled channels and orientation-dependent external interference, with the interference term vanishing under specified conditions.Source: 1) Epoch-Level RA Gradient · p. 8
• Projection onto the unit-sphere tangent space yields the Riemannian gradient.Source: 1) Epoch-Level RA Gradient · p. 8
• With neighboring boresights fixed, a local feasible set and linear oracle determine the update direction.Source: 2) Slew-Feasible Per-Epoch RA Update · p. 8

• The linear oracle selects the feasible candidate with the largest linear objective before the boresight is updated toward it.Source: I (AIAT · p. 8
• The normalized update preserves steering-cap and neighboring slew constraints under the stated parameter conditions.Source: I (AIAT · p. 8
• An Armijo search accepts a feasible step when the epoch oracle gap is positive.Source: I (AIAT · p. 8
• Odd and even epochs are updated in parallel, reducing serial depth from L updates to two color rounds while preserving feasibility and monotonic ascent.Source: 3) Two-Color Parallel RA-Trajectory… · p. 9
• The alternating algorithm repeatedly updates RA trajectories and serving sets until effective-throughput improvement falls below a prescribed tolerance.Source: D. Overall Algorithm, Convergence, and… · p. 9
• The integrated algorithm generates a feasible, nondecreasing effective-throughput sequence through surrogate maximization and Armijo-based RA updates.Source: D. Overall Algorithm, Convergence, and… · p. 9
• Both optimization blocks have polynomial per-iteration complexity and substantial epoch-level parallelism.Source: D. Overall Algorithm, Convergence, and… · p. 9

Simulation setup

• The default evaluation uses a 550-km Starlink Walker–Delta shell, a 3 × 3 UPA, Kmax = 6, 18.2 GHz carrier frequency, and 100 MHz bandwidth.The GS is located at (50°N, 120°E), and orbital parameters are randomized across realizations.Source: §IV · p. 9

Strength–separability tradeoff

• At ψ = 25°, throughput reaches 5.005 Gbps as effective rank rises to 4.72 despite normalized channel strength falling to 0.339.The intermediate optimum shows that spatial-separability gains can outweigh channel-strength loss.Source: A. Validation of the… · p. 10
• At ψ = 0°, co-directional channels yield rank one and 1.297 Gbps, while at ψ = 60° throughput falls to 3.646 Gbps after separation benefits diminish.The optimized-RA sweep therefore produces a unimodal throughput curve rather than favoring either extreme.Source: A. Validation of the… · p. 10

Algorithm validation

• Across 3 × 3, 4 × 4, and 5 × 5 UPAs, average throughput increases monotonically and stabilizes after approximately 15 iterations.The results are averaged over 200 orbital realizations under the default 10-dB reference INR, and larger arrays achieve higher throughput.Source: B. Validation of the Proposed Solution · p. 10

Loading and component contributions

• RA + MM consistently achieves the highest throughput, with satellite selection more influential when underloaded and RA reconfiguration increasingly important near full spatial loading.RA + Gain-TopK surpasses Fixed UPA + MM as Kmax approaches M, while the joint scheme benefits from both mechanisms.Source: B. Validation of the Proposed Solution · p. 11

Interference robustness

• Under stronger interference, RA + MM degrades more gradually and its gaps over single-component schemes widen because selection and RA shaping jointly account for interference exposure.With more interferers, interference becomes more diffuse and the gaps between joint and single-component schemes gradually narrow.Source: B. Validation of the Proposed Solution · pp. 11–12

Scheduling timescale

• A moderate number of scheduling epochs balances temporal adaptation against guard-interval overhead, whereas overly frequent updates reduce effective throughput.RA updates track changing geometry and interference, but additional guard intervals reduce payload duty factor.Source: B. Validation of the Proposed Solution · p. 12

Actuator limitations

• RA-enabled schemes improve monotonically with maximum angular velocity and largely saturate beyond approximately 20°/s.At low angular velocities, interference-aware selection can outperform slowly reconfigurable RAs paired with strength-only selection, showing that the mechanisms are complementary.Source: B. Validation of the Proposed Solution · pp. 12–13

• The two-timescale framework accounts for epoch-held serving sets and RA configurations, time-varying satellite geometry, exogenous cochannel interference, and inter-epoch slew constraints.Source: §V · p. 13
• Interference-whitened marginal-capacity analysis reveals that effective channel strength and spatial separability jointly shape multi-satellite reception.Source: §V · p. 13
• With fixed RA trajectories, the monotone submodular satellite-selection objective enables a discrete MM method based on an incumbent-tight modular lower-bound surrogate.Source: §V · p. 13
• Slew-feasible RA updates use Riemannian gradients and a two-color parallel update scheme, while the integrated alternating algorithm preserves feasibility and guarantees monotonic objective improvement and convergence.Source: §V · p. 13
• Simulations show consistent gains from joint design and an optimal balance between channel strength and spatial separability.Source: §V · p. 13