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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

TL;DR

The paper tackles finite-aperture design for fluid antenna arrays (FAA) by deriving a universal Cramér–Rao bound (CRB) that ties estimation accuracy to the geometric spread of port locations, and by obtaining a closed-form PDF for the minimum inter-port spacing under random placement. It then marries theory to practice with a gradient-based algorithm that optimizes continuous port positions within a fixed aperture, achieving substantial performance gains. Specifically, the CRB scales inversely with the geometric variance 𝓛_geo(p) = ∑(p_m − p̄)^2, and the minimum-spacing PDF enables principled constraint-setting; the optimization yields up to ~30% CRB reduction and ~42.5% AoA mean-squared error reduction across scenarios. Overall, the work provides universal FAA design insights and a practical offline optimization tool that outperforms conventional ULAs in finite-aperture settings.

Abstract

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.

Finite-Aperture Fluid Antenna Array Design: Analysis and Algorithm

TL;DR

The paper tackles finite-aperture design for fluid antenna arrays (FAA) by deriving a universal Cramér–Rao bound (CRB) that ties estimation accuracy to the geometric spread of port locations, and by obtaining a closed-form PDF for the minimum inter-port spacing under random placement. It then marries theory to practice with a gradient-based algorithm that optimizes continuous port positions within a fixed aperture, achieving substantial performance gains. Specifically, the CRB scales inversely with the geometric variance 𝓛_geo(p) = ∑(p_m − p̄)^2, and the minimum-spacing PDF enables principled constraint-setting; the optimization yields up to ~30% CRB reduction and ~42.5% AoA mean-squared error reduction across scenarios. Overall, the work provides universal FAA design insights and a practical offline optimization tool that outperforms conventional ULAs in finite-aperture settings.

Abstract

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 CRB reduction and a reduction in mean-squared error.
Paper Structure (16 sections, 29 equations, 5 figures, 1 algorithm)

This paper contains 16 sections, 29 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model
  3. Finite Aperture Configuration
  4. Signal Model
  5. Virtual Array Transformation
  6. Minimum Spacing Analysis
  7. Universal CRB for Finite-Aperture Array
  8. Derivative of the Steering Vector
  9. Expansion of the Quadratic Form
  10. Array Placement Algorithm Design
  11. Gradient Target Function Formulation
  12. Gradient Update Derivation
  13. Algorithm Design
  14. Computational Complexity
  15. Numerical Results
  16. ...and 1 more sections

Figures (5)

  • Figure 1: Empirical and theoretical $\mathbb E[\Delta_{\min}]$ of the minimum placement constraint $d_{\min}$: (a)$M\in\{3,4,\dots,16\}$, $W_{\max}=(M-1)\times 0.5$; (b)$M=8$, $W=10$.
  • Figure 2: Convergence behavior of Algorithm \ref{['alg:pgd']} under $M\in\{5,9,11\}$.
  • Figure 3: Illustration of different arrays for $M\in \{3,5,7,9,11\}$.
  • Figure 4: Maximum eigenvalue of the sensing codebook $\boldsymbol{A}$ and CRB of different arrays: (a)$M\in\{3,5,7,9,11\}$; (b)$\mathrm{SNR}=10$ dB, target angle $\theta=15^\circ$, snapshots $T=100$.
  • Figure 5: AoAMSE upper bound of different arrays: (a)$\mathrm{SNR}=10$ dB, $M\in\{3,5,7,9,11\}$; (b)$\mathrm{SNR}\in\{-10,-8,\ldots,20\}$ dB, $M=5$.