Polarization-Aware Rotatable Antennas for RIS-Empowered Symbiotic Radios

I. INTRODUCTION

• Antenna rotation can steer gain toward the DP-RIS and SR user, but also changes local polarization bases, so rotations, polarization states, and RIS phases require joint design.Independent continuous rotation of each antenna also motivates lower-complexity designs.Source: §I · pp. 1–2
• The proposed framework jointly optimizes digital beamforming, RA rotations, transceiver polarization states, and DP-RIS phase shifts under rate, interference-temperature, and hardware constraints.The alternating algorithm combines SDP, DC programming, and RCG methods.Source: §I · p. 2

II. SYSTEM MODEL

• The system has one SR user and K non-SR users, with M BS rotatable antennas and N DP-RIS elements arranged as UPAs.Each BS antenna has co-located H- and V-polarized ports; the RIS interleaves V- and H-polarized elements by column.Source: §II · p. 3

A. Rotation, Polarization, and DP-RIS Configuration

• The rotation model aligns the initial boresight, H-polarized, and V-polarized directions with the global x-, y-, and z-axes, respectively.The antenna rotation matrix determines the rotated directions.Source: A. Rotation, Polarization, and DP-RIS… · p. 3
• Polarforming reconfigures each BS antenna’s transmit polarization through complex weights at its dual-polarized ports, while the SR user has an adjustable receive polarization state.Non-SR users are modeled with fixed polarization antennas.Source: A. Rotation, Polarization, and DP-RIS… · p. 3
• The DP-RIS configuration assigns each reflecting element an H- or V-polarization direction and an independently adjustable phase shift.The phase shifts form a diagonal phase-shift matrix.Source: A. Rotation, Polarization, and DP-RIS… · p. 3

B. Channel Model

• The BS-to-DP-RIS channel uses a LoS path and multiple NLoS paths, with polarization coupling matrices modeling path polarization effects.The LoS path assumes ideal polarization preservation; NLoS coupling includes cross-polarization leakage and scatterer-induced random phases.Source: B. Channel Model · pp. 3–4
• Antenna rotation changes the boresight used in the directional large-scale fading model, whose peak directional gain depends on the antenna directivity factor.The model specifies path-dependent loss exponents and propagation directions for LoS and NLoS paths.Source: B. Channel Model · p. 3
• Small-scale fading uses orthonormal polarization bases perpendicular to the propagation directions and channel depolarization terms.The BS-to-RIS coefficients are collected into a channel matrix.Source: B. Channel Model · pp. 3–4
• The model forms direct BS-to-user and reflected RIS-to-user channel coefficients, then assembles them into user channel vectors.The unified indexing uses i = 0 for the SR user and i ≥ 1 for non-SR users.Source: B. Channel Model · p. 4

C. Transmission Model

• The primary rate is defined for each DP-RIS symbol state, while SIC and MRC over N_T residual samples yield processing gain N_T for decoding the secondary BPSK symbol.The DP-RIS symbol remains constant during one symbol period, with the secondary-to-primary symbol-period ratio assumed much greater than one.Source: C. Transmission Model · p. 4
• Average interference from both the direct BS signal and DP-RIS reflection at each non-SR user must remain below a predefined threshold.Source: C. Transmission Model · p. 4

III. TOY EXAMPLE: INSIGHTS ON RA DESIGN

• RAs steer radiation by changing boresight direction, unlike isotropic antennas, which radiate uniformly and cannot shape spatial radiation without beamforming.Source: §III · p. 4, Figure 2 · p. 4
• In the LoS toy setup, an RA steers its boresight toward the SR user, producing higher gain there and lower gain toward the non-SR user than a misaligned fixed directional antenna.The example places the SR and non-SR users at 35° and −20°, respectively; a larger directivity factor sharpens angular selectivity.Source: §III · p. 5, Figure 3 · p. 5
• Because useful spatial gain depends on alignment with a few dominant directions, shared rotations or a small direction-based codebook can reduce rotation complexity while preserving most spatial gain.Source: §III · p. 5

IV. PROBLEM FORMULATION

• The transmit-power minimization jointly optimizes BS rotations, digital beamforming, DP-RIS phases, and SR-user receive polarization under primary and secondary rate, interference, and hardware constraints.The formulation addresses coupled orientation, polarization, and reflection choices.Source: §IV · p. 5
• The constraints enforce primary rates for both DP-RIS symbol states, a minimum secondary rate, bounded average interference at non-SR users, and rotation and polarization hardware limits.Source: §IV · p. 5

V. PROPOSED SOLUTIONS

• An alternating-optimization framework updates the beamformer using SDP and DC programming, and optimizes rotations, polarization vectors, and RIS phases with RCG on their respective manifolds.Source: §V · p. 5

A. Optimization of the Digital Beamforming at the BS

• For the beamforming update, the matrix D = ww^H replaces the beamforming vector to handle non-convex quadratic terms in the rate and interference constraints.Source: A. Optimization of the Digital… · p. 5
• The beamforming subproblem’s rank-one constraint is handled with a DC penalty based on Tr(D) − λmax(D), linearizing the concave eigenvalue term at the previous iterate.Source: A. Optimization of the Digital… · p. 6
• The resulting P2.1 subproblem is convex and can be solved using CVX tools.Source: A. Optimization of the Digital… · p. 6

• The rotation update uses a max-min approach over normalized margins to find a feasible solution satisfying the rate, interference, and rotation-range constraints.Margins at or above one indicate satisfied rate requirements, while interference margins at or below one indicate the interference constraint is satisfied.Source: B. Optimization of the Rotation… · p. 6
• The rotation objective combines the worst margin with quadratic penalties for constraint violations, then smooths min and max operations using log-sum-exp and softplus approximations.Source: B. Optimization of the Rotation… · p. 6
• RCG optimizes the smoothed objective on the product manifold SO(3)^M, projecting gradients to tangent spaces and retracting updated rotations back onto the manifold.The method uses an Armijo backtracking line search for step sizes.Source: B. Optimization of the Rotation… · p. 6

• The transmit polarization vectors are optimized on a product of complex unit spheres using the same RCG framework as the rotation update.Source: C. Optimization of the Transceiver… · p. 6
• The transmit-vector update projects the Euclidean gradient onto each sphere’s tangent space, takes an RCG step, and retracts by normalization.Its Armijo backtracking rule sets the step size.Source: C. Optimization of the Transceiver… · pp. 6–7
• The receive polarization vector is constrained to a single complex unit sphere and uses the same RCG projection and normalization retraction.Source: C. Optimization of the Transceiver… · p. 7

D. Optimization of the DP-RIS Phase Shift Matrix

• The phase-shift matrix Θ is optimized on a complex circle manifold, with unit-modulus constraints on its diagonal entries.The subproblem is formulated with the other variables fixed.Source: D. Optimization of the DP-RIS Phase… · p. 7
• The Riemannian gradient is obtained by projecting the diagonal Euclidean gradient onto the complex circle manifold's tangent space.Only the diagonal entries of Θ are optimization variables.Source: D. Optimization of the DP-RIS Phase… · p. 7
• The RCG phase-shift update uses a retraction that normalizes each diagonal entry, and constraint violations trigger penalty-parameter increases.The algorithm increases η and λ_j upon violation before proceeding to the next variable.Source: D. Optimization of the DP-RIS Phase… · p. 7

1) Convergence Analysis:

• With the digital beamformer fixed, rotation, polarization, and DP-RIS updates maintain feasibility and improve normalized rate and interference-temperature constraint margins.These updates do not directly change transmit power; the beamforming update then minimizes it.Source: 1) Convergence Analysis · pp. 7–8
• The transmit-power sequence is non-increasing across AO iterations and converges, although the non-convex problem generally yields only a locally optimal solution.The updated beamformer minimizes power while the preceding feasible beamformer remains a feasible option.Source: 1) Convergence Analysis · p. 8

2) Computational Complexity:

• The stated overall complexity is I_AO(S_D + S_V + S_u + S_Θ), with the per-block costs depending on their respective DC or RCG iteration counts.The analysis separately gives update costs for the rotation matrices and for V, u_0, and Θ.Source: 2) Computational Complexity · p. 8

VI. LOW-COMPLEXITY DESIGNS FOR PRACTICAL RA IMPLEMENTATION

• For an array with M antennas, full 3D rotation control may require up to 3M actuators, motivating designs that reduce hardware and computation demands.The proposed alternatives are subarray-wise shared rotation and a discrete rotation codebook.Source: §VI · p. 8

A. Subarray-wise Shared Rotation Design

• The subarray-wise design assigns one common rotation matrix to every antenna in each of G disjoint subarrays.It reduces the rotation search space from SO(3)^M to SO(3)^G, where G ≪ M.Source: A. Subarray-wise Shared Rotation Design · p. 8
• The shared-rotation design retains continuous rotation optimization while reducing manifold variables and, with three rotation DoFs per antenna, cuts actuators from up to 3M to 3G.This offers a stated tradeoff between performance and complexity.Source: A. Subarray-wise Shared Rotation Design · pp. 8–9

B. Discrete RA Rotation Codebook

• The boresight codebook includes the SR-user direction, the DP-RIS direction, and intermediate directions between them.The intermediate directions are generated using a predefined set of weighting factors.Source: B. Discrete RA Rotation Codebook · p. 9
• With other variables fixed, each antenna's rotation is updated by discrete optimization over the predefined rotation codebook.The codebook is constructed from candidate boresight directions and deterministic rotation matrices.Source: B. Discrete RA Rotation Codebook · p. 9
• The discrete update requires M|C_R| objective evaluations per AO iteration instead of iterative RCG optimization over SO(3)^M, while preserving desired directional gain.The initial rotations should also come from the codebook to preserve the AO framework's monotonic convergence behavior.Source: B. Discrete RA Rotation Codebook · p. 9
• When antennas share rotations within subarrays, discrete search can run at the subarray level, reducing evaluations from M|C_R| to G|C_R|.The codebook design is compatible with the subarray-wise shared-rotation design.Source: B. Discrete RA Rotation Codebook · p. 10

VII. SIMULATIONS

• The simulations use a 4×4 BS array, a 4×8 DP-RIS, and two non-SR users under an interference-temperature constraint.The default settings include a maximum antenna rotation of π/4, p = 2, and an interference limit of −110 dBm.Source: §VII · p. 10

A. Performance Comparison with Benchmark Schemes

• All schemes except Baseline 1 monotonically reduce required transmit power and converge within roughly 10 iterations.Baseline 1 does not involve alternating optimization.Source: Figure 4 · p. 10, A. Performance Comparison with… · p. 10
• As the primary rate requirement increases, all schemes need more power and the proposed design remains lowest; Baselines 2 and 3 cross as the dominant constraint changes.At low primary rates, optimized DP-RIS phases help secondary transmission; at higher rates, primary-rate dominance makes BS polarization matching more important.Source: Figure 5 · p. 10, A. Performance Comparison with… · pp. 10–11
• The proposed joint design consistently achieves the minimum transmit power among the benchmark schemes as the secondary rate requirement increases.Optimizing DP-RIS phases gives passive beamforming gain, while polarforming further mitigates polarization mismatch.Source: Figure 6 · p. 11, A. Performance Comparison with… · p. 11
• The proposed design’s required transmit power decreases monotonically as directivity factor p grows, unlike fixed-orientation baselines, whose power eventually increases.With fixed boresights, narrower radiation patterns can worsen angular misalignment; rotation lets the proposed design use the increased directivity.Source: Figure 7 · p. 11, A. Performance Comparison with… · p. 11

B. Performance of Low-Complexity RA Designs

• When SR-user and DP-RIS directions are close, both low-complexity designs nearly match continuous rotation; their maximum reported gap is about 0.7 dBm for p = 2.As the directions separate, performance gaps grow, but remain limited; p = 5 requires less power than p = 2.Source: B. Performance of Low-Complexity RA… · pp. 11–12, Figure 8 · p. 12
• Increasing shared subarrays from one to two clearly reduces power, while further increases yield diminishing reductions and approach full continuous rotation performance.Two shared rotations can better support the direct SR and RIS-assisted cascaded links; more rotations add variables and complexity.Source: Figure 9 · p. 12, B. Performance of Low-Complexity RA… · pp. 12–13
• Adding candidates to the discrete rotation codebook lowers required power, but yields limited further improvement once the main spatial directions are represented.A small codebook can therefore retain most spatial gain while reducing rotation optimization complexity.Source: Figure 10 · p. 12, B. Performance of Low-Complexity RA… · p. 13

VIII. CONCLUSION

• The paper concludes that subarray-wise shared rotations and discrete codebooks preserve most full-continuous-design performance gain with reduced rotation complexity.The system jointly optimizes beamforming, antenna rotations, polarization states, and DP-RIS phases using an AO algorithm with SDP, DC programming, and RCG methods.Source: §VIII · p. 13