Finite-Aperture Modeling of Pattern-Reconfigurable Antennas for Multiuser Beamforming

I. INTRODUCTION

• Finite antenna size is absent from many ideal PRA models, which can admit unrealizable patterns and overestimate performance gains.The paper also identifies the lack of systematic validation between ideal and practical models and the complexity of jointly optimizing discrete switches and continuous beamforming.Source: §I · p. 2
• The proposed model links ideal aperture-based pattern constraints with practical hardware, and the optimization framework separates continuous Fourier coefficients from discrete switch states.The algorithm uses a coefficient-bound penalty-based WMMSE method and a one-bit neighborhood search that uses the Sherman-Morrison identity.Source: §I · p. 2

II. MODELING OF PRAS

• The finite-aperture ideal model uses a Fourier series and the Jacobi-Anger expansion to relate aperture size to effective Fourier order and coefficient power bounds.It provides an implementation-independent capability envelope for PRAs confined to a prescribed aperture.Source: §I · p. 2
• Practical reconfigurable pixel antennas are modeled with multiport networks, and their patterns are represented in the same Fourier domain as the ideal model.Three designs with different aperture sizes and pixel layouts exhibit aperture-dependent effective Fourier orders and satisfy the ideal model's per-order power constraints.Source: §I · p. 2, §II · p. 3

A. Introduction to PRAs

• Pattern reconfiguration changes path amplitudes and phases through the antenna's angular response, whereas position reconfiguration changes path phases through physical displacement.The two mechanisms therefore have different feasible sets of path weights.Source: A. Introduction to PRAs · p. 3

1) Order Constraint:

• For aperture radius R_A, coefficient bounds decay rapidly beyond |n| > eπR_A/λ, enabling a finite approximation that retains orders |n| ≤ N.This retains 2N + 1 Fourier orders, with N determined by the aperture’s electrical size.Source: 1) Order Constraint · pp. 3–4

2) Total Power Constraint:

• Normalizing angular power to unity yields a total-power constraint on the Fourier coefficients through orthogonality of the Fourier basis.Source: 2) Total Power Constraint · p. 4

3) Per-Order Power Constraint:

• Each Fourier order’s coefficient power is capped by min{1, C B_n}, where B_n is aperture-dependent and C characterizes admissible current intensity.Source: 3) Per-Order Power Constraint · p. 4
• Unlike prior ideal formulations,, the model uses complex Fourier coefficients to capture angular-response amplitude and phase, consistent with practical PRA designs.Source: 3) Per-Order Power Constraint · p. 4
• Aperture-linked truncation and order-dependent power bounds encode two physical conditions, yielding an aperture-dependent capability envelope for patterns achievable within a prescribed finite aperture.Source: 3) Per-Order Power Constraint · p. 4

• An RPA uses metallic pixels interconnected by RF switches; switching changes surface-current paths and reconfigures the radiation pattern.Source: C. Practical Model for PRAs · p. 4
• The network represents the feed and each PIN-diode connection as ports, with diode states modeled through switchable loads.Source: C. Practical Model for PRAs · pp. 4–5
• Switch-dependent pixel currents are solved from the network, and electromagnetic linearity combines feed and pixel embedded patterns weighted by their port currents.Source: C. Practical Model for PRAs · p. 5
• The practical model’s impedance parameters and open-circuit embedded radiation patterns can be obtained from full-wave electromagnetic simulations.Source: C. Practical Model for PRAs · p. 5

D. Consistency Between the Ideal and Practical Models

• The multiport network model’s normalized radiation patterns closely agree with full-wave electromagnetic simulations.This agreement supports using the network model to examine practical antennas’ Fourier-domain characteristics.Source: Figure 4 · p. 5, D. Consistency Between the Ideal and… · p. 5
• At retained order N=2 for RPA 1 and N=3 for RPA 2 and RPA 3, approximately 99.2% and 99.5% of pattern power is retained, respectively.The results support the aperture-dependent order constraint; RPA 2 and RPA 3 share an aperture and predicted order despite different pixel layouts.Source: D. Consistency Between the Ideal and… · p. 5, Figure 5 · p. 6
• All three designs’ realized Fourier-coefficient powers remain below theoretical per-order bounds, with a gap reflecting additional hardware constraints.At sufficiently high orders, coefficient powers become negligible; pixel geometry, switch topology, and realizable current distributions constrain practical antennas beyond the aperture-based envelope.Source: Figure 6 · p. 6, D. Consistency Between the Ideal and… · p. 6
• RPA 1 and RPA 2 yield denser, more diverse first-order coefficient samples than RPA 3, while RPA 2 and RPA 3 have different realizable regions despite sharing an aperture.The comparison separates aperture effects from hardware effects: the shared aperture gives RPA 2 and RPA 3 the same effective order, but their pixel layouts shape different coefficient regions.Source: Figure 7 · p. 6, D. Consistency Between the Ideal and… · p. 6
• The ideal model supplies an implementation-independent capability envelope, while the practical model describes patterns realizable by a specific antenna architecture.The aperture mainly sets effective Fourier order and the corresponding envelope; pixel layout and switch configuration determine the realizable coefficient region within it.Source: D. Consistency Between the Ideal and… · pp. 5–6

III. SYSTEM MODEL AND PROBLEM FORMULATION

• The downlink model uses a Fourier-domain channel to formulate weighted sum-rate maximization under both ideal and practical PRA models.The system has a base station with M PRAs serving K single-antenna users.Source: §III · p. 6, A. System Model · p. 6

A. System Model

• For practical PRAs, PIN-diode switch states determine multiport radiation patterns that are projected onto a truncated Fourier basis.The multiport model maps switch states to port currents and, through a coefficient mapping, to retained Fourier coefficients.Source: A. System Model · pp. 6–7
• The system incorporates channel-angle responses into a Saleh–Valenzuela channel model and expresses user signals under ideal or practical electromagnetic beamforming.The practical received-signal model replaces the ideal electromagnetic beamformer with the switch-state-dependent practical one.Source: A. System Model · p. 7

1) Ideal PRA Model:

• The ideal-model user SINR is defined from the received signal, and the K-user WSR aggregates users’ weighted rates.Each user’s weighting coefficient is denoted χk.Source: 1) Ideal PRA Model · p. 7
• The weighting coefficient χk specifies the weight assigned to user k in the WSR.The WSR is defined over K users.Source: 1) Ideal PRA Model · p. 7
• The ideal model constrains each antenna’s Fourier-coefficient vector γm to its feasible set.The constraint applies for m=1,…,M.Source: 1) Ideal PRA Model · p. 7

B. Problem Formulation

• The ideal formulation maximizes the K-user weighted sum rate subject to feasible Fourier coefficients and the transmit-power constraint.The user SINR and aggregate WSR are defined before stating the optimization problem.Source: 1) Ideal PRA Model · p. 7

1) Ideal PRA Model:

• The practical model replaces continuous Fourier coefficients with PIN-diode state vectors whose entries are binary.Its SINR and WSR use the practical electromagnetic beamformer Λ(t).Source: 2) Practical PRA Model · p. 7

2) Practical PRA Model:

• The ideal problem is the continuous counterpart of the practical formulation, with the discrete hardware mapping removed.The practical formulation retains the binary switch-state constraints.Source: 2) Practical PRA Model · p. 7

2) Practical PRA Model:

• The practical model imposes binary PIN-diode states, whereas the ideal problem is its continuous counterpart without the discrete hardware mapping.The proposed algorithm solves the practical mixed-binary formulation and uses its continuous component for the ideal problem.Source: 2) Practical PRA Model · p. 7

IV. CBP-WMMSE ALGORITHM

• CBP-WMMSE separates the mixed-binary design into continuous Fourier-coefficient optimization and discrete switch-state optimization.Its one-bit neighborhood search evaluates all single-bit neighbors and uses Sherman–Morrison updates for the loaded multiport impedance inverse.Source: §IV · p. 7

A. Problem Reformulation

• Auxiliary coefficients ηm are continuous surrogates for practical Fourier coefficients, with each order’s amplitude bounded by its hardware-achievable maximum ρn.The maximum amplitude is obtained by offline characterization over switch states.Source: A. Problem Reformulation · p. 8
• The reformulation uses the effective channel qk=Ξᴴhk and WMMSE equalizers and weights to handle the weighted-sum-rate objective.The MSE is defined for each user using its equalizer and weight.Source: A. Problem Reformulation · p. 8
• A penalty term incorporates the nonconvex equality between auxiliary Fourier coefficients and practical hardware coefficients into the objective.A phase rotation permits phase-invariant alignment, and the corresponding per-antenna phase can be absorbed into the digital beamformer without changing achievable system performance.Source: A. Problem Reformulation · p. 8
• The algorithm alternates variable-block updates and progressively increases the penalty coefficient until the hardware-consistency residual meets a prescribed tolerance.It starts with a small positive penalty to avoid the consistency term dominating the WMMSE objective too early.Source: A. Problem Reformulation · p. 8

B. Inner-Loop Variable Updates

• With other variables fixed, the equalizer block reduces to an optimization whose optimal user equalizers are given by the stated update.The excerpt identifies the block and its optimal equalizer expression.Source: B. Inner-Loop Variable Updates · p. 8
• With the equalizers fixed at their optimal values, each MSE weight is the reciprocal of that user’s MSE.The MSE is evaluated at the updated equalizer.Source: 2) Optimization of vk · p. 8
• The digital beamformer is optimized with the other variables fixed, using the effective-channel matrix and its Gram matrix in the subproblem.The excerpt defines Q from user effective channels and G=QᴴQ before isolating the terms that depend on W.Source: 3) Optimization of W · p. 8

W HQDQHW

• The Fourier-coefficient subproblem is optimized on a product of complex spheres with per-order amplitude bounds using projected Riemannian gradients.The projected tentative point uses Armijo backtracking and is accepted when it satisfies the sufficient-decrease condition.Source: W HQDQHW · p. 9
• For fixed Fourier coefficients, switch-state optimization separates across PRAs, and the phase rotation has a closed-form solution for each state.The phase rotation minimizes the residual for a given switch state.Source: 5) Optimization of tm and δm · p. 9
• OBNS evaluates all single-bit neighbors of the current switch state and selects the best neighbor, avoiding exhaustive evaluation of all 2^Q states.It accepts a flip only when the objective strictly improves; otherwise the search terminates.Source: 5) Optimization of tm and δm · p. 9
• The Sherman–Morrison identity updates the candidate inverse after a single switch flip, avoiding repeated matrix inversions during OBNS.A flip changes only one diagonal entry of the loaded-impedance matrix, and the current inverse is reused.Source: 5) Optimization of tm and δm · p. 9

C. Outer-Loop Updates and Stop Criteria

• For fixed penalty coefficient μ, the inner loop cycles through communication, Fourier-coefficient, and practical-state updates until the objective-change threshold or iteration limit is reached.The communication block updates user variables and beamforming; the practical-state block updates switch states and phase rotations.Source: C. Outer-Loop Updates and Stop Criteria · pp. 9–10
• The inner loop evaluates the residual used to decide whether the penalty coefficient should increase.The outer-loop criterion compares this residual against ε_out.Source: C. Outer-Loop Updates and Stop Criteria · pp. 9–10
• The outer loop increases μ by factor β when the hardware-consistency residual exceeds ε_out, stopping when the residual is at most ε_out or the outer-iteration limit is reached.The algorithm outputs the beamformer and practical switch states after normalization and phase rotation absorption.Source: C. Outer-Loop Updates and Stop Criteria · p. 10

D. Convergence Analysis

• For fixed penalty, each block update does not increase the objective, so the bounded-below inner-loop objective is non-increasing and converges.The outer loop successively increases the penalty, strengthening hardware consistency as the residual approaches zero.Source: D. Convergence Analysis · p. 10

E. Computational Complexity

• The outer loop’s penalty update has O(1) complexity per iteration; communication, beamformer, and Fourier-coefficient updates have the costs stated for their respective blocks.The communication-variable updates cost O(KMT + MK^2), and the beamformer update costs O(K^3 + MK^2).Source: E. Computational Complexity · p. 10
• Each OBNS round evaluates Q neighbors for each PRA and costs O(Q^2 + QT) per PRA; IF one-flip rounds across all PRAs cost O(IFM(Q^2 + QT)).Source: E. Computational Complexity · p. 10

F. Extensions

• The framework separates continuous beamforming from discrete hardware approximation, allowing the two parts to be adapted independently for different objectives and PRA implementations.The continuous part can use existing beamforming methods, while OBNS handles switch-state approximation.Source: F. Extensions · p. 10

V. SIMULATION RESULTS

• Ideal PRA models outperform practical PRA models, which outperform position-reconfigurable antennas and conventional arrays in sum rate across SNR.The ideal RPA 2 and RPA 3 models perform nearly identically because their aperture sizes match.Source: Figure 8 · p. 11, §V · p. 11
• The proposed algorithm achieves comparable sum rates to FP-Alt [35] and SEBO [27], with the highest RPA 3 sum rate throughout the tested antenna range.It also gives the best RPA 2 result at most tested antenna numbers.Source: Figure 9 · p. 11, §V · p. 11
• For RPA 2 with M = 60, the proposed algorithm requires approximately 1/12 of SEBO’s runtime, while its runtime grows only moderately as switch count increases.For RPA 3 with Q = 12, it is slower than FP-Alt and SEBO because continuous Fourier-coefficient optimization adds overhead.Source: Figure 10 · p. 11, §V · p. 11
• At 1000 evaluations, OBNS attains a lower objective than SEBO and exhaustive search, using approximately 1/15 and 1/60 of their runtimes, respectively.It reuses the current inverse and intermediate quantities rather than repeating matrix inversions.Source: Figure 11 · p. 11, §V · p. 12

VI. CONCLUSION

• The paper combines finite-aperture ideal modeling, practical multiport models, WSR optimization, and efficient switch-state search for PRA-enabled multiuser communications.Its conclusion reports sum-rate gains over position-reconfigurable and conventional arrays and reduced runtime for densely switched RPAs.Source: §VI · p. 12

APPENDIX A PROOF OF PROPOSITION 1

• Finite-aperture radiation has Bessel-governed angular modes and a rapidly decaying envelope on sufficiently high-order Fourier coefficients.The Jacobi-Anger expansion relates the aperture radiation integral to the Fourier representation.Source: Appendix A · p. 12

APPENDIX B DERIVATION OF ρn

• Radiation-power normalization alone does not strictly bound current norm because non-radiating components and destructive cancellation can preserve normalized far-field power.The paper therefore introduces a separate current-capability constraint.Source: Appendix B · p. 13
• The current-capability constraint PW ≤ C yields the per-order bound |γ_n|² ≤ CB_n, where B_n depends on aperture geometry.A smaller C represents more restrictive current intensity and weaker high-order excitation.Source: Appendix B · pp. 12–13
• The total power constraint also implies |γ_n|² ≤ 1 for each Fourier coefficient.Source: Appendix B · p. 13