Rotatable Antenna-Enabled Wireless Communication: Modeling and Optimization

• Conventional large-scale MIMO increases array gains but also hardware costs and power consumption, while fixed antennas cannot adjust their positions or orientations.Source: §I · p. 1
• RAs independently adjust each directional antenna’s 3D orientation or boresight through mechanical or electronic means without translational movement.This local rotational adjustment is presented as compatible with existing wireless systems.Source: §I · p. 2
• The paper introduces pointing vectors for RA orientations and incorporates boresight rotation into multipath geometric-based channel models, including polarization and frequency-selective fading extensions.Source: §I · p. 2
• For single-user free-space propagation with MRC, the authors derive closed-form optimal RA pointing vectors and SNR expressions and bounds for ULA and UPA settings.The resulting SNR first increases linearly with antenna number and eventually converges to a limit.Source: §I · p. 2
• For multi-user multipath channels, the receive beamformer and RA pointing vectors are jointly optimized for minimum SINR using an iterative AO algorithm and a non-iterative two-stage algorithm.The two-stage method reduces computational complexity by using weighted channel power gain maximization with ZF beamforming.Source: §I · p. 2
• Simulations validate the analysis and show significant performance improvements over benchmark schemes, including gains with small rotational ranges and stronger benefits under higher antenna directivity.Source: §I · p. 2

Directional antenna model

• Each RA uses a directional gain pattern determined by signal angles relative to its boresight and a directivity factor controlling main-lobe beamwidth.The gain is maximal in the boresight direction under the adopted power-conserving pattern.Source: B. Channel Model · pp. 3–4

Multipath channel

• The narrowband frequency-flat model represents line-of-sight and non-line-of-sight components using geometric near-field propagation through distributed scatterer clusters.The overall multipath channel is formed by superimposing the LoS and NLoS components.Source: B. Channel Model · p. 4

Propagation gains

• Channel power gains account for path loss and directional gain, with user and scatterer directions obtained from RA, user, and cluster positions.The model uses projected direction vectors between each RA boresight and the relevant propagation direction.Source: B. Channel Model · p. 4

Model extensions

• The geometric model can incorporate standardized path loss, polarization effects, and frequency-selective fading for broader channel settings.The main exposition uses a narrowband frequency-flat model for convenience, while extensions are described elsewhere in the paper.Source: B. Channel Model · p. 4

Uplink reception

• The received uplink signal includes user transmit signals, linear receive beamforming, and additive white Gaussian noise.The receive beamforming vectors are normalized before extracting each user’s signal and forming its SINR.Source: B. Channel Model · p. 4

Problem formulation

• The optimization jointly selects the receive beamforming matrix and RA pointing matrix to maximize the minimum SINR among users.The pointing vectors must satisfy the allowed rotational range and unit-vector constraints.Source: C. Min-SINR Maximization Problem · p. 4

Channel acquisition

• Channel estimation uses path loss, AoA/AoD, delay, and pre-measured directional patterns to construct effective channels for candidate boresight directions.This avoids estimating separate channels for every possible boresight orientation.Source: C. Min-SINR Maximization Problem · pp. 4–5

Imperfect CSI

• With CSI estimation errors, the resulting SINR expression serves as a performance lower bound while retaining the form used by the proposed analysis and algorithms.The estimation error is incorporated into the effective noise term through the effective transmit SNR.Source: C. Min-SINR Maximization Problem · p. 5

Single-user solution

• In the single-user free-space case, MRC is optimal for any fixed pointing matrix, and the pointing problem decomposes into independent subproblems for individual RAs.Each subproblem maximizes the projection between an RA pointing vector and the user-direction vector.Source: §III · p. 5, A. Optimal Closed-Form Solution · p. 5
• The closed-form optimum points every RA toward the user whenever the rotational constraint permits that alignment, achieving maximum directional gain.The resulting maximum directional gain is NG0 when all boresights align with the user direction.Source: A. Optimal Closed-Form Solution · p. 5

Analysis setup

• The single-user analysis derives a closed-form ULA SNR, UPA bounds, and asymptotic gains as antenna number increases.The user is placed on the z-axis for the main geometric analysis.Source: B. Asymptotic Performance Analysis · p. 5

Geometric regions

• The array divides into inner and outer areas: inner RAs can align boresights with the user, whereas outer RAs remain offset by the zenith-angle constraint.The offset is determined by the maximum allowable rotational angle.Source: B. Asymptotic Performance Analysis · p. 5

ULA scaling

• When the ULA span remains within the rotational range, the maximum SNR increases linearly with the number of RAs.This result follows from the closed-form analysis under the cosine directional gain pattern.Source: 1) ULA-Based RA System · p. 6
• After the ULA span exceeds the rotational range, the SNR growth rate gradually decreases and eventually approaches zero as more RAs are added.The saturation reflects the increasing contribution of outer-array elements whose boresights cannot fully align with the user.Source: 1) ULA-Based RA System · p. 6

Asymptotic comparison

• A larger allowable boresight rotation range produces higher asymptotic SNR and a larger performance gap over fixed antennas.The comparison follows from exploiting boresight rotation as an additional spatial degree of freedom.Source: 1) ULA-Based RA System · p. 6

General user location

• For users around the ULA, all geometric cases converge to the same asymptotic SNR as the array length tends to infinity.The limiting value is independent of the user’s azimuth angle in this asymptotic regime.Source: 1) ULA-Based RA System · p. 6

Inner-area regime

• When the entire UPA lies within the inner area, the optimal SNR increases linearly with the antenna number N = NxNy.The condition limits the UPA diameter relative to propagation distance and rotational range.Source: 2) UPA-Based RA System · pp. 6–7

Large-array regime

• For large UPAs exceeding the inner area, a closed-form SNR is difficult because the expression contains a double integral and a circular inner–outer boundary.The paper therefore derives lower and upper bounds instead.Source: 2) UPA-Based RA System · p. 7

Bound construction

• The UPA region is bounded using inscribed and circumscribed disks whose radii determine the lower and upper SNR bounds.The inner-area radius is limited by the smaller UPA dimension, while the outer radius follows from the diagonal.Source: 2) UPA-Based RA System · p. 7

Asymptotic behavior

• As the UPA dimensions grow without bound, the lower and upper SNR bounds approach the same limit.Both bounds use the same limiting form of the function G(R, p, θmax) as the disk radius tends to infinity.Source: 2) UPA-Based RA System · p. 7

• The multi-user problem maximizes the minimum SINR to balance directional gains among users.Source: §IV · p. 7
• The receive beamforming and RA pointing vectors are intricately coupled, making the objective non-concave and motivating alternating optimization.Source: A. AO Algorithm · p. 7
• For a fixed RA pointing matrix, the channel becomes fixed and the problem reduces to a receive-beamforming subproblem.Source: 1) Receive Beamforming Optimization · p. 7
• Each user’s receive SINR is a generalized Rayleigh quotient, so MMSE beamforming maximizes the receive SINR for fixed pointing vectors.Source: 1) Receive Beamforming Optimization · p. 7
• The beamforming implementation uses interference-plus-noise covariance matrices and a Woodbury identity to reduce matrix-inversion dimension from N × N to (K − 1) × (K − 1).Source: 1) Receive Beamforming Optimization · p. 7

• With receive beamforming fixed, a slack variable η represents the minimum SINR in the reformulated optimization problem.Source: 2) RA Pointing Vector Optimization · p. 7
• The pointing-vector subproblem remains nonconvex because the unit-norm constraint is difficult to handle directly.Source: 2) RA Pointing Vector Optimization · p. 8
• The multipath channel is rewritten per RA, yielding the composite channel vector hk(F) used in the optimization.Source: 2) RA Pointing Vector Optimization · p. 8
• Taking logarithms gives an equivalent SINR constraint, but the channel term remains neither convex nor concave because of complex path coefficients.Source: 2) RA Pointing Vector Optimization · p. 8
• Successive convex approximation uses first-order Taylor expansions to replace the difficult constraint with a convex approximation and obtain a local solution iteratively.Source: 2) RA Pointing Vector Optimization · p. 8
• Relaxing ∥⃗fn∥ = 1 to ∥⃗fn∥ ≤ 1 produces a convex problem solvable with CVX, whose optimum is an upper bound for the unre laxed subproblem.Source: 2) RA Pointing Vector Optimization · p. 8

• The AO algorithm partitions the variables into receive beamforming V and RA pointing matrix F, then optimizes the two blocks alternately.Source: 3) Overall Algorithm · p. 8
• Each AO iteration uses the MMSE beamformer and solves the relaxed pointing-vector problem, with complexity driven by matrix inversion and CVX optimization.Source: 3) Overall Algorithm · pp. 8–9
• The AO objective value is nondecreasing across iterations because each block update optimizes its corresponding subproblem.Source: 3) Overall Algorithm · p. 9
• The two-stage algorithm avoids iteration by optimizing RA pointing vectors with semidefinite relaxation and then obtaining beamforming with zero forcing.Source: B. Two-Stage Algorithm · p. 9
• Zero-forcing beamforming requires N ≥ K, removes inter-user interference, and reduces each user’s SINR to an SNR.Source: B. Two-Stage Algorithm · p. 9
• Under zero forcing, the resulting SNR mainly depends on the channel power gain ∥¯hk∥2, with ρZF,k representing cancellation-related SNR loss.Source: B. Two-Stage Algorithm · p. 9

1) First stage:

• Semidefinite relaxation converts the nonconvex rank-one pointing-vector problem into a convex semidefinite program solvable by CVX.The relaxed solution may require rank-one approximation through eigenvalue decomposition to construct a feasible pointing vector.Source: 1) First stage · pp. 9–10
• The two-stage algorithm reconstructs channels and then calculates zero-forcing beamforming after obtaining the stacked pointing vector.Its complexity is dominated by the semidefinite program, singular value decomposition, and one-time beamforming calculation.Source: 2) Second Stage · p. 10
• The two-stage method has lower complexity than AO because it solves the optimization and beamforming steps only once.Its lower complexity comes with possible performance loss from setting p = 1, whereas AO supports arbitrary p.Source: 2) Second Stage · p. 10
• For ULA systems, RA received power increases linearly with antenna count before approaching an asymptote, achieving up to 1.43 dB over fixed antennas and up to 5 dB when Nx ≤100.The closed-form SNR matches the exact value, while the RA system reaches its asymptotic limit later.Source: Figure 5 · p. 10, A. Single-User System · p. 10
• Increasing directivity raises RA received power, while fixed-antenna power first increases and then decreases as the main lobe narrows.The RA system benefits from adjusting its directional gain toward the user.Source: A. Single-User System · pp. 10–11, Figure 6 · p. 11
• RA maintains more uniform received power across user azimuth angles than fixed antennas by reconfiguring its directional gain pattern.Fixed-antenna power decreases sharply away from the region directly in front of the array.Source: Figure 7 · p. 11, A. Single-User System · p. 11

B. Multi-User Narrow-Band System

• The AO algorithm’s max-min achievable rate increases over iterations and converges within six iterations.The evaluated benchmarks include random orientation, fixed orientation, and isotropic-antenna designs.Source: B. Multi-User Narrow-Band System · p. 11
• AO with MMSE beamforming achieves the highest max-min achievable rate, while the two-stage algorithm still provides up to 5 dB over fixed antennas.The algorithms therefore offer different performance-complexity trade-offs.Source: B. Multi-User Narrow-Band System · pp. 11–12
• Increasing θmax gives optimized RA more flexibility to balance directional gain and increase max-min achievable rate.The optimized RA system consistently outperforms fixed antennas, while random orientations decline when θmax ≥3π/10.Source: B. Multi-User Narrow-Band System · p. 12
• Even a small rotational range can yield substantial performance gains in the proposed RA system.The growth rate of its max-min communication rate rises sharply when θmax ≤π/10.Source: B. Multi-User Narrow-Band System · p. 12

C. Polarization-Aware Wideband Channel

• In polarization-aware wideband OFDM, RA sum-rate versus directivity is unimodal, improving up to around p = 5 and then degrading.The polarization-aware RA system consistently outperforms the other schemes across the tested directivity range.Source: C. Polarization-Aware Wideband Channel · pp. 12–13
• All schemes gain sum-rate as the number of subcarriers increases, while polarization-unaware RA remains slightly below polarization-aware RA.More subcarriers reduce relative cyclic-prefix overhead and increase allocation flexibility for RA systems.Source: C. Polarization-Aware Wideband Channel · p. 13, Figure 13 · p. 13

APPENDIX A PROOF OF THEOREM 1

• When the array dimensions satisfy the stated geometric bound, the SNR expression reduces before evaluating the resulting double integral.The proof then applies linear arctangent approximations and integral identities to obtain Lemma 2.Source: Appendix A · p. 14, Appendix B · p. 14

APPENDIX C PROOF OF THEOREM 2

• Theorem 2 introduces D = min{R, r tan θmax} to capture the effective radial extent constrained by distance and boresight rotation.This quantity appears in the decomposition of the approximated SNR expression.Source: Appendix C · p. 14
• The proof evaluates the first and second double integrals separately before substituting both results into the theorem expressions.The derivation uses the stated integral calculations and the p = 1 condition.Source: Appendix C · p. 14, Appendix D · p. 14
• The final substitution yields Theorem 2 under the condition relating R, D, and θmax.The proof uses θmax = arctan(D/r) when R ≥ D.Source: Appendix D · p. 14

Polarization-aware channel model

• Antenna rotation changes the polarization axis, potentially causing transmit–receive polarization misalignment and mismatch loss.The model assumes linearly polarized antennas at both the BS and users.Source: Appendix E · p. 14
• The rotated polarization direction is represented from the RA rotation angles and projected onto the tangent plane orthogonal to each propagation direction.Separate projections are used for user-to-RA and scatterer-cluster-to-RA links.Source: Appendix E · p. 15
• Polarization-aware channel gains are incorporated for user-to-RA and cluster-to-RA links, updating the LoS and NLoS channel gains.The electric-field vectors of transmitted and reflected waves enter the corresponding channel-gain expressions.Source: Appendix E · p. 15

Wideband OFDM channel model

• The continuous-time channel is transformed into a spatial-frequency response and then into the discrete frequency-domain OFDM channel across L subcarriers.With subcarrier spacing ∆f, total bandwidth is B = L∆f and the l-th subcarrier frequency is f_l = (l − 1)∆f.Source: Appendix E · p. 15

Wideband OFDM transmission

• Each subcarrier is assigned to at most one user, with binary allocation variables and per-subcarrier transmit-power constraints.Perfect synchronization and AWGN are assumed for the received signal model.Source: Appendix E · p. 15

Optimization

• Under single-user subcarrier transmission, MRC maximizes receive SNR, and the resulting rate is used to optimize RA pointing and subcarrier allocation for sum rate.The pointing matrix and allocation variables are alternately optimized; the allocation subproblem is binary integer programming, while the pointing subproblem is handled through convex approximation and conditional gradients.Source: Appendix E · pp. 15–16