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Reconfiguring Sparse Apertures: Partial-Mobility Planar FAS for 2-D DOA Estimation

Xiaokai Song, Tuo Wu, Jie Tang, Maged Elkashlan, K. C. Ho, Kin-Fai Tong

arXiv:2609.06375v1eess.SP

TL;DR

Fixed sparse geometries trade wide-sector identifiability against local angular resolution. This paper uses a fixed Nyquist seed for acquisition and partially reconfigures a planar FAS for refinement; equal-budget simulations report lower angular error and much of the all-movable benefit with fewer actuated ports.

  • Problem

    Fixed sparse arrays can improve local sensitivity but risk ambiguity, while compact arrays support wider-sector search at the cost of aperture for resolving nearby sources.

    Source: §I · p. 1

  • Method

    A fixed Nyquist seed selects an angular basin, after which selected ports reconfigure and the seed and refinement observations are jointly processed using policy-aware information analysis and exact-FIM design.

    Source: §I · p. 2, A. FAS Reconfiguration State · p. 3

  • Results

    Equal-budget simulations report lower angular error from reconfiguration and retention of much of the all-movable benefit with fewer actuated ports.

    Source: §VII · p. 13

  • Takeaways & Limitations

    Partial mobility offers an accuracy–cost tradeoff, but a permanently deployed full sparse aperture can be preferable when movement and snapshot splitting are avoidable.

    Source: A. Seeded Reconfiguration Protocol · p. 7

    A fully movable FAS defines the information upper bound, and fixing seed ports is not claimed to give a lower CRB than moving all ports.

    Source: A. Seeded Reconfiguration Protocol · p. 7

Abstract

from arXiv · show

Classical sparse arrays enlarge the sensing aperture under limited element and radio-frequency (RF) chain budgets, but their fixed geometries impose a persistent tradeoff between wide-sector identifiability and local angular resolution. This paper converts this static design into a two-state reconfigurable sparse aperture by allowing only part of a planar fluid antenna system (FAS) to move after an initial observation. A compact, Nyquist-spaced seed first provides an ambiguity-controlled acquisition geometry. The remaining fluid ports then move or switch to information-efficient refinement positions, and the measurements collected before and after reconfiguration are jointly processed. We formulate the receiver under fixed total numbers of ports and RF chains, a movement budget, and a common total-snapshot budget. A policy-level Fisher information matrix (FIM) identity accounts for the observation-dependent refinement geometry. For a single source, a relaxed planar D-optimal analysis gives the corner-favoring aperture law, while a finite-port certificate bounds the information retained under spacing, reachability, and partial-actuation constraints. For multiple sources, an aperture-and-conditioning surrogate generates feasible sparse layouts that are ranked and locally refined using the exact frame FIM with coarray, spacing, and movement regularization. Seed multiple signal classification (MUSIC) identifies a reliable angular basin, followed by joint concentrated-likelihood refinement over both states. Equal-budget simulations show that reconfiguring a sparse aperture converts available area and movement into lower angular error and retains much of the all-movable gain with fewer actuated ports. They also identify the operating boundary: if a full sparse aperture can remain permanently deployed, avoiding movement and snapshot splitting can be preferable.

I. INTRODUCTION

Fixed sparse arrays trade wide-sector identifiability for local angular resolution, motivating a receiver that uses a compact seed for acquisition and partially movable ports for refinement. The paper develops this protocol under movement, RF, and snapshot constraints and evaluates its information and estimation methods.

I. INTRODUCTION

  • A compact half-wavelength array supports wide-sector search, while long sparse baselines improve local sensitivity but can produce ambiguity and narrow attraction basins.A fixed geometry cannot reallocate sensors after observing the scene.Source: §I · p. 1
  • The proposed protocol keeps a calibrated Nyquist seed fixed for acquisition, then moves or switches selected fluid ports to form source-conditioned sparse baselines for refinement.Measurements from both states are retained; moving every port would increase actuation, switching, recalibration, and settling overhead.Source: §I · p. 2
  • The method derives a policy-level FIM identity, a relaxed rectangular-aperture D-optimal bound, and finite-port information-retention certificates under spacing, reachability, and actuator constraints.For multiple sources, an aperture-and-conditioning surrogate generates candidates that exact-FIM ranking and local refinement assess.Source: §I · p. 2
  • Seed MUSIC selects a reliable angular basin before joint concentrated-ML refinement, while equal-budget experiments compare reconfigured designs with compact, frozen, random-reconfigured, and all-movable baselines.The evaluation varies area, movement, port count, source separation, prior shift, and position error.Source: §I · p. 2

A. FAS Reconfiguration State

  • The receiver uses a compact half-wavelength seed patch and movable or switchable free ports within the same planar aperture, with settling and a fixed total snapshot budget.The model connects the seed ports first and all ports after reconfiguration; its main setup uses N simultaneous receive chains.Source: §I · p. 2, A. FAS Reconfiguration State · p. 3

B. Spatial-Frequency Observation Model

The observation model represents source directions by spatial frequencies and combines seed and refinement measurements. It distinguishes information for a realized layout from the policy-level information of a data-dependent reconfiguration.

B. Spatial-Frequency Observation Model

  • The model represents each source by spatial frequencies u_l = sin ϑ_l cos ϕ_l and v_l = sin ϑ_l sin ϕ_l, with steering vectors evaluated at calibrated port coordinates.The source samples are unknown deterministic nuisance parameters, and the total snapshot budget is T_tot = T_0 + T_1.Source: B. Spatial-Frequency Observation Model · p. 3
  • The joint observation comprises seed data and refinement data whose layout is selected by a policy from the seed observation, R(1) = π(X(0)).The seed term stabilizes basin selection, while the refinement term supplies most of the high-resolution information.Source: B. Spatial-Frequency Observation Model · p. 3
  • Under the stated regularity and policy assumptions, the joint equivalent FIM adds the seed-state and conditional refinement-state information after state-specific waveform nuisance parameters are eliminated.The conditional refinement score has zero conditional mean given the seed data, so the cross terms vanish.Source: B. Spatial-Frequency Observation Model · pp. 3–4
  • A realized seed observation gives a conditional layout-information diagnostic, whereas averaging over seed observations yields the policy FIM that characterizes the adaptive rule before acquisition.True-direction placement is an oracle benchmark; an implementable controller uses a seed confidence region, posterior, or slow-time prior.Source: B. Spatial-Frequency Observation Model · p. 4

C. 2-D Difference Coarray

The difference coarray maps physical port pairs to virtual-array lags, including a Nyquist patch suitable for ambiguity-controlled processing. A FAS policy differs from a frozen array because its reachable refinement geometry can depend on the seed observation.

C. 2-D Difference Coarray

  • The 2-D difference coarray indexes virtual-array entries by physical position differences, and a rectangular Nyquist patch at d0 = λ/2 supports ambiguity-controlled subspace processing.Because continuous locations rarely yield exact lattice lags, the design rewards differences near the desired patch instead of requiring exact equality.Source: C. 2-D Difference Coarray · p. 4

D. Frozen Arrays and FAS Reconfiguration Policies

  • A FAS policy selects a reachable refinement geometry after the seed observation, whereas a frozen array fixes its coordinates before acquisition.Setting B_mv = 0 or selecting the refinement geometry independently of seed data recovers a frozen design.Source: D. Frozen Arrays and FAS… · p. 4
  • The numerical baselines include compact UPA, frozen sparse-grid, random-reconfigured, frozen continuous, and all-movable designs, with estimators separating geometry gains from joint multistate-processing gains.The FAS feasible set is state dependent because reconfiguration follows coarse directional information.Source: D. Frozen Arrays and FAS… · p. 4

E. Performance Metrics

The section defines angle-domain CRB and RMSE evaluation, a conditional FIM consistent with concentrated ML, and single-source aperture benchmarks with a separate finite-port certificate.

E. Performance Metrics

  • The angular CRB and permutation-matched RMSE are evaluated in physical azimuth/polar angles, with azimuth errors wrapped before source assignment.The two-state CRB uses the same total snapshot budget as the corresponding RMSE experiment.Source: E. Performance Metrics · p. 4

A. Conditional FIM Consistent With the ML Estimator

  • The conditional FIM removes unknown deterministic waveform nuisance parameters by orthogonal projection and averages information using the empirical waveform covariance.For the uncorrelated equal-power simulations, the covariance is normalized to P = I_L.Source: A. Conditional FIM Consistent With the… · pp. 4–5

B. Single-Source Closed Form

  • The relaxed single-source D-optimal design favors equal mass at the rectangle corners, while finite-port certificates bound retention for reachable, spacing-feasible layouts.The asymptotic (Nf/N)^2 factor is determinant retention in the relaxed model, not a direct CRB ratio or finite-port guarantee.Source: B. Single-Source Closed Form · pp. 5–6

C. Multi-Source Conditioning

For multiple sources, aperture size alone does not ensure identifiability: steering correlation can make the estimation problem ill-conditioned. The design therefore balances coordinate spread against source separation and conditioning.

C. Multi-Source Conditioning

  • Large coordinate covariance can still yield poor identifiability when nearby steering and derivative subspaces become nearly collinear.A small minimum eigenvalue of the normalized steering Gram matrix signals this conditioning problem.Source: C. Multi-Source Conditioning · p. 6
  • For two equal-power sources, the normalized Gram eigenvalues are 1 ± |γ(∆η)|, so useful aperture growth also requires suppressing correlation over relevant separations..Source: C. Multi-Source Conditioning · p. 6
  • The aperture-conditioning tradeoff can pull free ports from corners toward edges or the interior to reduce average correlation over the design sector.Source: C. Multi-Source Conditioning · p. 6

D. Global-Basin Control for Large Apertures

Large apertures improve local estimation sensitivity but complicate global search. A half-wavelength seed provides an ambiguity-controlled basin, after which larger nonuniform baselines support local refinement.

D. Global-Basin Control for Large Apertures

  • Large port separations provide high Fisher information but narrow main lobes and add sidelobe structure that can make unrestricted MUSIC or ML initialization fragile.The fixed seed avoids exact visible-sector aliases in the ideal manifold, though finite-snapshot perturbations and peak merging remain possible.Source: D. Global-Basin Control for Large… · p. 6
  • The half-wavelength seed patch preserves an ambiguity-free scan region over the open visible disk, while its compact aperture limits final accuracy.The seed selects a local basin and its likelihood contribution is retained during joint refinement.Source: D. Global-Basin Control for Large… · p. 6
  • After seed-based basin selection, larger nonuniform baselines can reduce the local CRB without requiring a global large-aperture search.Source: D. Global-Basin Control for Large… · p. 6

IV. MOVEMENT-CONSTRAINED FAS RECONFIGURATION AND ESTIMATION

The receiver keeps a compact Nyquist seed fixed while moving only the remaining ports after seed observation. It designs and evaluates refinement layouts under information, coarray, spacing, and movement considerations.

A. Seeded Reconfiguration Protocol

  • The fixed seed supplies an ambiguity-controlled reference for coarse MUSIC and basin selection, while moved free ports provide coordinate covariance for final precision.Both states contribute to joint refinement; fixing seed ports is a hardware-aware accuracy–cost tradeoff, not a claim of lower CRB than an unconstrained all-movable array.Source: A. Seeded Reconfiguration Protocol · pp. 6–7

B. FIM-Informed D-Optimal Movement-Regularized Reconfiguration

  • The movement-regularized criterion jointly rewards frame information and local lag support while penalizing spacing violations and port movement.With zero movement budget the geometry is frozen.Source: B. FIM-Informed D-Optimal… · p. 7
  • Aperture-and-correlation candidates are generated with a smooth coarray coverage regularizer for local MUSIC candidate generation.The logarithmic soft count rewards lag coverage without repeatedly rewarding the same lag.Source: B. FIM-Informed D-Optimal… · p. 7
  • Candidates are ranked and locally refined using the exact multistate objective, and returned layouts must pass movement and spacing checks.Reported layouts and CRBs use the matched FIM rather than the surrogate.Source: B. FIM-Informed D-Optimal… · p. 7

C. Gradient Structure and Feasibility Handling

The exact local refinement uses differentiable FIM terms with box and movement constraints, while spacing is penalized during search and checked afterward. Its KKT guarantee is conditional and local, not global.

C. Gradient Structure and Feasibility Handling

  • Only the refinement-state FIM depends on the design, and analytic derivatives can remove the Nf multiplier.Finite differences are used for candidate generation and exact-objective refinement; slowly varying layouts can be cached.Source: C. Gradient Structure and Feasibility… · pp. 7–8
  • Box and movement constraints are imposed by the solver, while spacing is penalized during search and checked afterward.Vanishing projected-gradient or KKT residuals establish only local first-order stationarity because the problem remains nonconvex.Source: C. Gradient Structure and Feasibility… · p. 7
  • Under constant-rank, noncoincidence, and constraint-qualification assumptions, exact local refinement converges to a first-order KKT point of the constrained problem.The result assumes a line-search SQP or projected method with sufficient ascent and exact or asymptotically accurate finite-difference derivatives.Source: C. Gradient Structure and Feasibility… · p. 7
  • The multi-start surrogate supplies candidate initializations but provides no global guarantee.The stationarity result concerns local refinement only.Source: C. Gradient Structure and Feasibility… · p. 8
  • Finite-difference refinement costs KIs[QNL2 + N 2|G0|] + KQ[NL2 + L3], using the stated surrogate and exact-FIM evaluation costs.K is the number of starts, Is and Ie are surrogate and exact iterations, and |G0| is the number of target lags.Source: C. Gradient Structure and Feasibility… · p. 8

D. Seeded FAS-ML Estimator

The seeded estimator uses confidence-gated seed MUSIC to choose whether to move free ports, then combines both states in local concentrated-likelihood refinement. Practical robustness depends on stable directions and manageable movement and settling time.

D. Seeded FAS-ML Estimator

  • Seed MUSIC supplies coarse directions and a confidence score; movement is skipped below a threshold and otherwise free ports move to refinement positions.The gate uses the weakest selected peak relative to the median seed-spectrum level, so movement is triggered only when all selected peaks are sufficiently prominent.Source: D. Seeded FAS-ML Estimator · p. 8
  • Seed peaks gate local neighborhoods, refinement-state MUSIC proposes candidates, and concentrated ML jointly refines both states using the additive information model.The seed suppresses remote basins while the moved aperture supplies fine differential-phase information.Source: D. Seeded FAS-ML Estimator · p. 8
  • The estimator replaces exhaustive multi-source grid enumeration with a small candidate set and local refinement; slowly varying deployments can retrieve layouts from an offline sector codebook.After covariance formation, each two-state likelihood evaluation costs O(N 2L + NL2 + L3).Source: D. Seeded FAS-ML Estimator · p. 8

V. ROBUSTNESS AND PRACTICAL CONSTRAINTS

  • The DOAs must remain approximately constant across seed acquisition, reconfiguration, and refinement; if latency is limiting, the controller can use an offline sector codebook.Movement and settling time contribute to the angular-coherence requirement.Source: §V · p. 8
  • Position errors cause direction-dependent coherence loss, while geometry-dependent coupling can be handled with a calibrated response in the FIM and likelihood.The design objective can be averaged over position-error samples to introduce robustness.Source: §V · p. 8

VI. SIMULATION RESULTS

The simulations examine information gain, estimator performance, actuation tradeoffs, and source-prior changes under a fixed total snapshot budget. The figures show how the compact seed and moved aperture provide complementary basin-search and local-resolution behavior.

VI. SIMULATION RESULTS

  • The default study uses N = 16 ports, a 3 × 3 seed, seven free ports, and a square deployment region of side D = 24d0.The FAS splits the fixed snapshot budget evenly between seed and refinement; frozen arrays use all snapshots at one geometry.Source: §VI · pp. 8–9

A. Reconfiguration and Basin Formation

  • Free ports move toward corners and edges while calibrated seed ports remain compact, enlarging coordinate covariance and strengthening dominant FIM terms.The movement pattern and rounded post-movement 2-D difference coarray illustrate the proposed refinement mechanism.Source: A. Reconfiguration and Basin Formation · p. 9, Figure 2 · p. 9
  • Post-movement MUSIC has narrow peaks at true directions but remote sidelobes, whereas seed MUSIC forms a broader connected coarse basin.The two states are complementary; the seed gate matters more when peaks merge or finite-snapshot perturbations make initialization unreliable.Source: A. Reconfiguration and Basin Formation · p. 9, Figure 3 · p. 9

B. Aperture and Movement Scaling

Reconfiguration turns deployment area and movement into lower CRB, while the fixed seed preserves acquisition reliability and free-port design determines refinement quality.

B. Aperture and Movement Scaling

  • As deployment side length D grows, reconfigured FAS CRB decreases, while compact UPA and frozen sparse-grid CRBs remain nearly unchanged.This is an information-theoretic result; practical value depends on seed-controlled refinement.Source: B. Aperture and Movement Scaling · p. 9, Figure 4 · p. 9
  • As normalized movement budget Bmv/Bfull increases, prior-averaged CRB, test-pair CRB, and seeded FAS-ML RMSE all decrease.Empirical RMSE tracks test-pair CRB, while prior-averaged CRB measures performance beyond the selected pair.Source: B. Aperture and Movement Scaling · p. 10, Figure 5 · p. 10

C. Estimation Accuracy and Resolution

D. Partial Actuation and Design Ablations

  • With the 3 × 3 seed fixed, optimized free-port layouts achieve the lowest CRB; the first movable ports add especially valuable long nonredundant baselines.Later ports mainly improve conditioning and robustness.Source: D. Partial Actuation and Design… · p. 10, Figure 7 · p. 11
  • CRB decreases as more ports move, and the numerically optimized all-movable design is lowest in the comparison but requires every port to be actuated.The plotted all-movable point is a feasible benchmark, not the information upper bound.Source: D. Partial Actuation and Design… · p. 10, Figure 8 · p. 11
  • The 3 × 3 seed is an intermediate resolution choice: it sharply improves near the target SNR while leaving seven ports for refinement movement.The 4 × 4 seed reaches the resolution threshold at lower SNR but consumes all 16 ports.Source: D. Partial Actuation and Design… · p. 10
  • The regularized placement sacrifices a small amount of D-optimal determinant gain to retain local rounded-Nyquist-lag support, which alone does not predict estimator RMSE.The final estimator uses the exact reconfigured-array manifold.Source: D. Partial Actuation and Design… · pp. 10–11

E. Optimization Behavior and Computational Cost

Seeded estimation keeps search overhead close to the seed scan as grids refine, while robustness tests show that gains depend on source geometry and deployment conditions.

E. Optimization Behavior and Computational Cost

F. Robustness and Sector Coverage

G. Prior Adaptation

  • Optimized free-port layouts depend on the prior: broad sectors favor corner and edge diversity, while low-elevation or close-source priors favor aligned baselines..Source: G. Prior Adaptation · p. 12
  • When a well-conditioned full aperture can remain deployed, avoiding movement and snapshot splitting outweighs adaptation in the reported comparison.All frozen-to-reconfigured ratios remain below one under the three prior shifts.Source: G. Prior Adaptation · p. 12, Figure 14 · p. 12

VII. CONCLUSION

  • Under equal snapshot budgets, reconfiguration reduces angular error and retains much of the all-movable benefit with fewer actuated ports, but its benefit is limited when a full aperture can remain deployed.The paper identifies compact standby, retractable deployment, or limited actuation as its most relevant settings.Source: §VII · p. 13
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