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Finite-Aperture Fluid Antenna Array Design: Analysis and Algorithm

Zhentian Zhang, Kai-Kit Wong, Hao Jiang, Farshad Rostami Ghadi, Hyundong Shin, Yangyang Zhang

arXiv:2601.18471v1cs.IT

TL;DR

Finite-aperture constraints limit the reliability of classical sparse geometries and leave FAA design rules under fixed apertures insufficiently established. The paper derives closed-form spacing and CRB analyses, then uses them to design continuous port locations; the optimized FAA reduces CRB by about 30% and AoAMSE by 42.5%.

  • Problem

    Finite-aperture constraints complicate array design, while fundamental limits and design rules for FAA placement remain missing.

  • Method

    The paper derives a closed-form CRB and minimum-spacing PDF, then proposes a gradient-based algorithm for optimizing continuous FAA port locations.

  • Results

    About 30% CRB reduction at M = 11 and 42.5% average AoAMSE reduction for M = 5 are achieved compared with the ULA and across SNR regimes, respectively.

  • Takeaways & Limitations

    Closed-form geometric analysis and optimization provide finite-aperture FAA design guidance, with numerical results showing improved performance over conventional ULAs.

Abstract

from arXiv · show

Finite-aperture constraints render array design nontrivial and can undermine the effectiveness of classical sparse geometries. This letter provides universal guidance for fluid antenna array (FAA) design under a fixed aperture. We derive a closed-form Cramér--Rao bound (CRB) that unifies conventional and reconfigurable arrays by explicitly linking the Fisher information to the geometric variance of port locations. We further obtain a closed-form probability density function of the minimum spacing under random FAA placement, which yields a principled lower bound for the minimum-spacing constraint. Building upon these analytical insights, we then propose a gradient-based algorithm to optimize continuous port locations. Utilizing a simple gradient update design, the optimized FAA can achieve about a $30\%$ CRB reduction and a $42.5\%$ reduction in mean-squared error.

I. INTRODUCTION

Finite-aperture constraints make classical sparse array designs less reliable, motivating fluid antenna arrays as a flexible architecture. The paper develops analytical guidance and a practical optimization approach for FAA port placement.

  • Motivation: Finite-aperture constraints complicate array design because existing sparse geometries are not necessarily effective when the aperture is limited.Port placement affects estimation precision and sidelobe behavior.
  • Motivation: Fluid antenna systems treat antennas as reconfigurable physical-layer resources that provide an additional design degree of freedom.FAS has been applied to beamforming, sensing, and multiple access.
  • Research gap: FAA design under finite-aperture constraints lacks fundamental limits and design rules despite prior work providing an upper bound for finite-aperture AoA sensing.The paper addresses this missing guidance under fixed aperture budgets.
  • Contributions: The paper derives a closed-form CRB that captures different finite-aperture geometries and unifies reconfigurable array analyses.The analysis is intended to clarify geometric limits under a fixed array length.
  • Contributions: A closed-form PDF characterizes minimum port spacing under random placement, providing an analytical basis for the minimum-spacing constraint.The spacing analysis is paired with a gradient-based algorithm for practical FAA design.
  • Finite-aperture configuration: The FAA model fixes the first and last ports at the aperture boundaries while optimizing intermediate continuous port locations subject to minimum spacing.The normalized aperture is Wmax = (M −1)/2, and adjacent ports must satisfy pm −pm−1 ≥ dmin.

C. Virtual Array Transformation

The paper transforms covariance measurements into a virtual difference co-array determined by pairwise port differences, then derives random-placement minimum-spacing statistics and a principled spacing bound.

  • Virtual Array Transformation: Covariance vectorization produces a virtual received signal whose virtual difference co-array is determined by the set of port-position differences.The sensing codebook is formed by discretizing the angular domain and constructing one column per angle from the array steering vector.
  • Minimum Spacing Analysis: Randomly placed ports are modeled as uniformly distributed order statistics over the fixed aperture, with minimum spacing defined between adjacent sorted positions.The joint density is uniform over the ordered simplex, enabling analysis of the minimum-spacing random variable.
  • Minimum Spacing Analysis: The analysis motivates setting the FAA minimum-spacing constraint no smaller than the derived expected minimum spacing.The paper states this recommendation as Proposition 1 after deriving the minimum-spacing PDF.
  • Minimum Spacing Analysis: The closed-form minimum-spacing distribution is obtained by converting exclusion-zone constraints into a reduced-aperture standard simplex through volume-preserving coordinate transformations.The transformed effective aperture is W′ = Wmax − (M−1)δ, and differentiating the resulting CDF yields the PDF.
  • Minimum Spacing Analysis: The ULA maximizes minimum spacing, whereas random FAA placement has an expected minimum spacing that decays quadratically with the number of ports.ULA spacing decreases as ∝M−1, while expected random-FAA minimum spacing decreases as ∝M−2.

IV. UNIVERSAL CRB FOR FINITE-APERTURE ARRAY

This section derives a finite-aperture CRB by connecting Fisher information to antenna-position geometry under a deterministic signal model.

  • IV. UNIVERSAL CRB FOR FINITE-APERTURE ARRAY: The closed-form CRB explicitly relates angle-estimation accuracy to antenna-position variance and applies across antenna placements under a finite aperture.The derivation provides a theoretical benchmark for different systems and placement geometries.
  • IV. UNIVERSAL CRB FOR FINITE-APERTURE ARRAY: The steering vector uses wavenumber k = 2π/λ, with positions normalized by wavelength so the steering component becomes e−j2πpm cos θ.After normalization, the derivation treats antenna positions as deterministic constants.
  • IV. UNIVERSAL CRB FOR FINITE-APERTURE ARRAY: The Fisher information element Jθθ is derived with the Slepian–Bangs formula under the deterministic signal model.The derivation uses the real part operator, signal power, the steering-vector derivative, and a projection onto the null space of the steering vector.

A. Derivative of the Steering Vector

The steering-vector derivative exposes antenna positions directly through a diagonal position matrix, providing the geometric term used in the CRB derivation.

  • A. Derivative of the Steering Vector: Differentiating the m-th steering-vector element with respect to angle yields a factor proportional to jkpm sin(θ) times the element itself.The derivative follows from [a(θ)]m = e−j kpm cos(θ).
  • A. Derivative of the Steering Vector: In vector form, the derivative is ˙a(θ) = jk sin(θ)Da(θ), where D is diagonal with the antenna positions.This compact representation makes the dependence of the derivative on port placement explicit.

B. Expansion of the Quadratic Form

The derivation reduces the Fisher-information term to the geometric variance of port locations, yielding a closed-form CRB that exposes how aperture geometry controls estimation precision.

  • B. Expansion of the Quadratic Form: The quadratic form is reduced using the antenna-position derivative and inner-product expansions into a term involving the array’s geometric centroid and variance.The geometric centroid is introduced through the port positions, and the variance identity converts the resulting expression into effective aperture variance.
  • B. Expansion of the Quadratic Form: The Fisher-information element becomes proportional to sin^2(θ)L_geo(p), linking estimation information directly to geometric variance.The proportionality includes observation duration, SNR, and the squared wavenumber.
  • B. Expansion of the Quadratic Form: The closed-form CRB is inversely proportional to L_geo(p), so increasing geometric variance improves estimation precision.For wavelength-normalized positions, the CRB is written as 1/[8π^2T·SNR·sin^2(θ)·L_geo(p)].
  • B. Expansion of the Quadratic Form: Maximizing geometric variance by placing elements near aperture boundaries minimizes CRB but can induce false spectral peaks and beam misdirection.Uniform spacing suppresses these false peaks but provides a smaller effective aperture and lower precision.

V. ARRAY PLACEMENT ALGORITHM DESIGN

The placement formulation converts the AoAMSE objective into a geometry-aware optimization that balances estimation precision against ambiguity suppression under finite aperture.

  • V. ARRAY PLACEMENT ALGORITHM DESIGN: The finite-aperture design objective is to optimize port placement using the ratio γmax(p)/λ̄2(p), based on the AoAMSE upper bound.Here, γmax(p) is the largest eigenvalue of A^H(p)A(p), while λ̄2(p) is the effective aperture term.
  • V. ARRAY PLACEMENT ALGORITHM DESIGN: Logarithm conversion yields an AoAMSE minimization expression separating the eigenvalue and effective-aperture contributions.The formulation uses the logarithms of the two terms to construct the placement objective.
  • V. ARRAY PLACEMENT ALGORITHM DESIGN: The effective aperture term is directly proportional to geometric variance, so minimizing −ln(λ̄2) is equivalent to minimizing the CRB.The geometric-variance relation follows from Lagrange’s identity.
  • V. ARRAY PLACEMENT ALGORITHM DESIGN: Minimizing ln(γmax) suppresses sidelobes, making the objective seek a Pareto-optimal balance between estimation precision and ambiguity resolution.The section then motivates a gradient-based algorithm for solving the resulting placement problem.

A. Gradient Target Function Formulation

The target function combines effective-aperture maximization with correlation control through the dominant eigenvalue of the sensing Gram matrix.

  • A. Gradient Target Function Formulation: The Gram matrix is defined as Q(p)=A^H(p)A(p), with γmax(p) its dominant eigenvalue and λ̄2(p) the effective squared-aperture term.These quantities provide the two geometry-dependent components used in the optimization objective.
  • A. Gradient Target Function Formulation: The target function is formulated from the logarithmic eigenvalue and effective-aperture terms introduced by the placement objective.It is based on the joint precision–ambiguity formulation in (26).

B. Gradient Update Derivation

The algorithm derives position gradients for aperture spread and correlation reduction, then applies projected gradient descent while preserving fixed boundaries and minimum spacing.

  • B. Gradient Update Derivation: The gradient with respect to each intermediate port position contains two derivative terms corresponding to the effective-aperture and dominant-eigenvalue components.Only positions p2 through pM−1 are differentiated and updated.
  • B. Gradient Update Derivation: The effective-aperture derivative pushes antenna elements away from the array centroid to maximize spatial spread.This term increases the geometric separation that contributes to the effective aperture.
  • B. Gradient Update Derivation: The dominant-eigenvalue derivative adjusts positions to minimize correlation among the sensing responses.This complements aperture expansion by targeting ambiguity-related structure.
  • B. Gradient Update Derivation: Projected gradient descent updates the intermediate ports and restores feasibility through sorting, edge pinning, and forward/backward minimum-spacing corrections.The algorithm is initialization-sensitive but is stated to guarantee a local minimum value.

D. Computational Complexity

The evaluation compares placement schemes and shows that the proposed algorithm converges, while optimized FAA designs improve CRB and AoAMSE under finite apertures.

  • D. Computational Complexity: O(N^3 + M^2N^2) per iteration is dominated by Gram-matrix eigen-decomposition and matrix construction with gradient updates.The eigenvalue evaluation contributes O(N^3), while construction and gradient updates contribute O(M^2N^2).
  • D. Computational Complexity: The experiments compare discrete FAS, continuous FAS, and other antenna placements under normalized wavelength and fixed-aperture settings.Continuous FAS is initialized from scaled MRA positions and optimized using Algorithm 1.
  • D. Computational Complexity: Monte Carlo samples agree with the predicted minimum-spacing distributions, validating the analytical placement constraint.Figure 1 evaluates empirical and theoretical expected minimum spacing across antenna counts and aperture settings.
  • D. Computational Complexity: Algorithm 1 converges under multiple antenna-count settings, although the small step size requires several hundred iterations.The optimized objective is significantly lower than that of the conventional ULA, and the offline design requires only one-time computation.
  • D. Computational Complexity: About 30% CRB reduction is observed at M = 11 for FAA relative to ULA, with continuous and discrete FAA showing similar CRB performance.The performance gap between FAA and ULA widens as M increases.
  • D. Computational Complexity: 42.5% average AoAMSE reduction is achieved for M = 5 across different SNR regimes.The AoAMSE gap between FAS and FAA becomes more pronounced as aperture size increases.

VII. CONCLUSION

The paper develops analytical tools and an optimization algorithm for finite-aperture FAA design. Numerical results show reductions in both CRB and AoAMSE relative to conventional ULA-based design.

  • VII. CONCLUSION: The paper derives a closed-form PDF for the expected minimum placement gap and a closed-form CRB applicable to general finite-aperture array designs.The CRB is explicitly related to antenna-position variance, while the minimum-spacing distribution characterizes finite-aperture placement limits.
  • VII. CONCLUSION: The proposed practical optimization algorithm substantially reduces both CRB and AoAMSE under a fixed aperture.The numerical evaluation compares maximum sensing-codebook eigenvalues and CRB across different arrays, and evaluates AoAMSE upper bounds.
  • VII. CONCLUSION: The numerical results confirm the superiority of fluid antennas over conventional ULAs for finite-aperture array design.The conclusion summarizes the reported advantage across the paper's finite-aperture evaluations.
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