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Flexible Beamforming via Fluid Antenna Array: From Discrete to Continuous Array
Jingyuan Xu, Yuanhui Wu, Zhentian Zhang, Zaichen Zhang, Tuo Wu, Hao Jiang, Jian Dang, Christos Masouros, Kin-Fai Tong
TL;DR
Planar FAA beam synthesis must select a small number of ports while matching a desired far-field pattern, despite phase ambiguity and the difficulty of continuous position optimization. The paper combines phase-retrieval-assisted discrete selection with continuous joint refinement. In the evaluated two-beam setup, the complete method improves mainlobe energy fraction and peak sidelobe level relative to a matched fixed array.
Problem
Sparse FAA synthesis must choose a small active subset from a dense grid, while greedy selection is sensitive to response correlation and amplitude-only targets lack phase.
Source: A. Background and Related Work · p. 2Method
FFT-based phase retrieval constructs an aperture-adapted complex target, OMP and least squares initialize a discrete solution, and projected Adam jointly refines continuous positions and weights with spacing repair.
Source: C. Contributions · p. 2, §III · p. 4Results
48.92% versus 35.73% mainlobe energy fraction and −3.83 dB versus −2.30 dB PSLL: the complete method improves both metrics over the fixed array with the same aperture and active-port count.
Source: §V · p. 11Takeaways & Limitations
In the evaluated two-beam setup, the complete FAA design achieves higher mainlobe concentration and lower sidelobes than the matched fixed-array baseline.
Source: §V · p. 11
Abstract
from arXiv · showhide
Fluid antenna arrays (FAAs) reconfigure their effective aperture by activating radiation ports from a dense set of candidate positions within a finite physical region, thereby offering geometry diversity beyond conventional fixed arrays. This paper studies arbitrary-direction far-field beam synthesis with a planar FAA. Since a fixed array adapts its beamforming weights on a predetermined spatial sampling structure, its ability to match desired beam patterns is limited when both the aperture and the number of active elements are constrained. To exploit the geometry diversity of the FAA, we propose a coarse-to-fine scheme that couples discrete port selection with continuous joint refinement. In the coarse stage, a fast Fourier transform (FFT) based phase retrieval step converts the amplitude-only beam specification into an aperture-compatible complex target, after which an FFT-tailored greedy method selects the active ports and least squares initializes their complex weights. In the fine stage, a projected adaptive moment estimation procedure jointly refines the continuous port positions and beamforming weights while enforcing the minimum interelement spacing. For the evaluated two-beam configuration, the proposed scheme increases the mainlobe energy fraction from 35.73% to 48.92% and reduces the peak sidelobe level from -2.30 dB to -3.83 dB relative to a fixed array with the same aperture and active-port count.
I. INTRODUCTION
Fluid antenna arrays can adapt their active-port geometry as well as their weights, offering additional freedom for far-field beam synthesis within a fixed aperture and port budget. The paper addresses phase ambiguity and difficult off-grid optimization with a coarse-to-fine design, then formulates the planar far-field model on a spatial-frequency grid.
A. Background and Related Work
- Multiple active ports make the FAA layout part of the array manifold, so jointly adapting port locations and weights adds geometry diversity beyond fixed-layout weight adaptation.The paper distinguishes this deterministic structural flexibility from fading diversity.Source: A. Background and Related Work · pp. 1–2
B. Challenges and Motivations
- Amplitude-only specifications leave target phase undetermined, while greedy selection and continuous position optimization face correlation, nonconvexity, grid-resolution, and spacing challenges.The proposed coarse-to-fine design uses discrete selection to initialize continuous joint refinement.Source: A. Background and Related Work · p. 2, B. Challenges and Motivations · p. 2
C. Contributions
- FFT-based Gerchberg–Saxton phase retrieval constructs an aperture-adapted complex target before port selection.This step addresses the amplitude-only specification's missing phase.Source: C. Contributions · p. 2
- FFT-tailored OMP selects candidate ports and least squares initializes weights before Adam jointly refines continuous coordinates and weights with spacing repair and a beam-balancing loss.The feasible iterate with the highest mainlobe concentration is retained.Source: C. Contributions · pp. 2–3, Figure 2 · p. 3
A. Far-Field Beam Model for the FAA
- The far-field model maps directions to (u,v), samples the visible disk on a uniform FFT grid, and evaluates synthesis under a shared aperture, port count, and desired beam.The visible hemisphere is represented by the disk in spatial-frequency coordinates.Source: A. Far-Field Beam Model for the FAA · p. 3
B. Discrete Port-Selection Formulation
The discrete formulation represents a planar aperture with a dense candidate-port grid and selects exactly S active ports to approximate the desired beam while respecting the minimum inter-port distance.
B. Discrete Port-Selection Formulation
- The aperture is covered by an Mc × Nc rectangular candidate grid with Np = McNc ports and spacing dcand.Grid coordinates are vectorized in column-major order.Source: B. Discrete Port-Selection Formulation · pp. 3–4
- A steering dictionary stores each candidate port's far-field response, and a weight vector synthesizes the beam over the sampled directions.Nonzero weights identify active ports and their complex beamforming weights.Source: B. Discrete Port-Selection Formulation · p. 4
- Port selection minimizes beam-approximation error with exactly S nonzero weights, subject to the prescribed minimum inter-port distance.When dcand ≥ dmin, every subset of distinct grid points is feasible for the discrete initialization.Source: B. Discrete Port-Selection Formulation · p. 4
C. Continuous Joint Design Formulation
The formulation relaxes the candidate-grid restriction by allowing active ports to move continuously within the same physical aperture, subject to aperture and spacing constraints. It jointly designs port positions and complex weights, using the discrete OMP solution to initialize refinement.
C. Continuous Joint Design Formulation
- Each active port position is represented by r_s = [x_s, y_s]^T, with both coordinates constrained to the physical aperture.The coordinate bounds are 0 ≤ x_s ≤ W_x and 0 ≤ y_s ≤ W_y.Source: C. Continuous Joint Design Formulation · p. 4
- The active ports must satisfy a prescribed minimum inter-port distance.The spacing constraint applies between distinct active ports.Source: C. Continuous Joint Design Formulation · p. 4
- For a given placement, the position-dependent steering matrix represents the active-port responses used in beam synthesis.The matrix is denoted D(R) and has dimensions Z × S.Source: C. Continuous Joint Design Formulation · p. 4
- Each active port has a complex beamforming weight whose magnitude and phase specify its excitation amplitude and phase.The active-port weights are collected in w.Source: C. Continuous Joint Design Formulation · p. 4
- Continuous positions and complex weights are jointly optimized for beam synthesis under aperture and minimum-spacing constraints.The model permits each active port to take any continuous position in the aperture, rather than restricting it to a candidate grid.Source: C. Continuous Joint Design Formulation · p. 4
- Directly solving the continuous formulation is challenging because port coordinates affect the steering matrix nonlinearly and spacing constraints create a nonconvex feasible set.The method replaces the squared-error objective with the performance-oriented loss and initializes refinement from OMP-selected positions and least-squares weights.Source: C. Continuous Joint Design Formulation · p. 4
A. Array–Beam Fourier Relation
The array factor and aperture weights are connected by a two-dimensional Fourier relation, enabling FFT-based conversion between beam and aperture representations. OMP then selects a sparse set of candidate ports for a complex target, followed by least-squares weight fitting.
A. Array–Beam Fourier Relation
- Under the far-field model, the planar-aperture array factor is related to the complex aperture weights through a two-dimensional spatial Fourier transform.On the reciprocal grid, the relation reduces to a two-dimensional DFT, up to indexing, centering, and normalization conventions.Source: A. Array–Beam Fourier Relation · p. 4
- An inverse DFT implemented with an IFFT maps the complex beam-domain matrix to a sampled aperture distribution.The Fourier relation applies to uniformly sampled aperture weights and their reciprocal beam-domain samples.Source: A. Array–Beam Fourier Relation · p. 5
1) OMP Port Selection:
- OMP selects S positions from N_p candidates to approximate the complex target beam with a sparse weight vector.The sparse approximation uses the antenna steering dictionary and complex beamforming weights.Source: 1) OMP Port Selection · p. 5
- At each OMP iteration, the method selects the unchosen atom with maximum absolute residual correlation, refits by least squares, and updates the residual.After S selections, it recomputes weights using the unnormalized selected dictionary; the selected positions and weights initialize continuous refinement.Source: 1) OMP Port Selection · p. 5
2) Phase Retrieval:
Because an amplitude mask leaves phase unspecified, the method uses Gerchberg–Saxton alternating projections to construct a complex target compatible with finite aperture support. FFT and IFFT updates alternate between beam and aperture domains before the target is passed to OMP.
2) Phase Retrieval:
- An amplitude-only beam specification leaves phase unconstrained, and an arbitrary phase can yield a complex target difficult to approximate with finite aperture.The method uses Gerchberg–Saxton alternating projections to seek a phase more compatible with the aperture support.Source: 2) Phase Retrieval · p. 5
- The phase-retrieval iteration starts from a broadside-referenced linear phase and maps the current beam to the aperture domain with an IFFT.The initial phase uses a slope k referenced to the grid indices corresponding to u = v = 0.Source: 2) Phase Retrieval · p. 5
- Each iteration truncates the aperture distribution to the finite L × L support, transforms it back to the beam domain, and restores the prescribed visible-disk amplitude.Outside the visible disk, the intermediate beam-domain field is retained.Source: 2) Phase Retrieval · p. 6
- After T_PR alternating projections, visible-domain entries retain the specified amplitude and carry the retrieved phase, forming the vectorized OMP target.Each iteration uses one two-dimensional IFFT and one two-dimensional FFT over the P × Q grid, with complexity O(T_PR P Q log(PQ)).Source: 2) Phase Retrieval · p. 6
C. Continuous Stage: Projected Adam Joint Refinement
Projected Adam refines the OMP initialization by jointly optimizing continuous port coordinates and complex weights. Its loss promotes mainlobe concentration and balances concentration across desired beams, while spacing repair maintains minimum separation.
C. Continuous Stage: Projected Adam Joint Refinement
- Projected Adam jointly refines the selected port coordinates and their complex beamforming weights to exploit the continuous aperture.The OMP solution supplies the grid-constrained initial solution.Source: C. Continuous Stage: Projected Adam… · p. 6
1) Optimization Variables and Initialization:
- The optimization variables are the port-position matrix and the real and imaginary parts of the complex weights, initialized from OMP positions and least-squares weights.Adam updates the weight components and port coordinates jointly.Source: 1) Optimization Variables and… · p. 6
2) Loss Function:
- The loss promotes concentration of response energy within prescribed mainlobe regions and balances concentration across desired beams.The mainlobe energy fraction is evaluated on a uniform (u, v) grid inside the visible disk; the logarithmic concentration objective is invariant to common amplitude scaling.Source: 2) Loss Function · p. 6
- After every position update, spacing repair enforces the minimum interelement distance, and peak sidelobe level provides a complementary evaluation of the strongest sidelobe.Fitting the target supplies initialization, while the refinement promotes mainlobe concentration and balance among beams.Source: 2) Loss Function · p. 6
3) Adam Optimization:
Projected Adam jointly refines port positions and complex weights, while projecting positions into the aperture and repairing minimum-spacing violations. The method retains the best eligible feasible checkpoint, or the feasible OMP initialization if none qualifies.
3) Adam Optimization:
- The position, real-weight, and imaginary-weight blocks share one loss, use independent Adam moment states, and update jointly from gradients evaluated at the same state.Automatic differentiation returns the three gradients in one pass.Source: 3) Adam Optimization · pp. 6–7
- After each position update, coordinates are projected into the aperture and repaired for minimum spacing; the repair also runs once more for final feasibility verification.The repair uses iterative pairwise repulsion and clips the result back to the aperture.Source: 3) Adam Optimization · p. 7
- After each weight update, the complex weights are rescaled to their initial norm because the loss depends only on power ratios.Source: 3) Adam Optimization · p. 7
- The method returns the eligible feasible checkpoint with the highest mainlobe concentration, retaining the feasible OMP initialization if no Adam iterate qualifies.Eligibility requires finite quantities, feasible spacing, and an inter-beam concentration gap within the prescribed threshold.Source: 3) Adam Optimization · p. 7
- The stated total cost is O(ZS^2 + TAdam[ZS + TprojS^2]), including the initial least-squares fit, Adam field and gradient evaluation, and spacing repair.Source: 3) Adam Optimization · p. 7
A. Simulation Setup
The simulations compare fixed and fluid arrays under shared aperture and active-port constraints, using a two-beam target and common visible-disk sampling. The setup specifies the target shape, evaluation samples, and refinement settings.
A. Simulation Setup
- The fixed and fluid arrays use the same physical aperture and active-port count, while fluid ports are selected from a denser candidate grid.This setting isolates port-selection freedom under a shared aperture and hardware budget.Source: A. Simulation Setup · p. 8
- The target comprises two flat-top beams with linear roll-off, sharing an elevation center and using distinct azimuth centers.Each beam has a 3.6° flat-top half-width and a 4.4° linear-transition half-width.Source: A. Simulation Setup · p. 8, Figure 3 · p. 8
- The evaluation uses all visible-disk samples, with mainlobe and sidelobe sets defined by target-amplitude thresholds of 0.5 and 0.05, respectively.The mainlobe energy fraction is reported as a percentage; three-dimensional patterns use a 181 × 181 display grid.Source: A. Simulation Setup · p. 8
- Adam-refined results use cosine decay for position learning rate, warm-up followed by cosine decay for weight learning rate, and a fixed initial weight norm.The reported checkpoints must satisfy minimum spacing and the prescribed inter-beam concentration-gap threshold.Source: A. Simulation Setup · p. 9
B. Discrete-Stage Comparison
The discrete-stage comparison evaluates phase retrieval and fluid port selection before continuous refinement. Fluid+GS gives the strongest reported mainlobe concentration and lower PSLL than the fixed baseline.
B. Discrete-Stage Comparison
- The discrete comparison holds the 4.5λ × 4.5λ aperture and S = 100 active ports constant while varying fixed or fluid placement and phase retrieval.Fluid selects ports from a denser candidate grid; all configurations use least squares for weights after positions are fixed.Source: B. Discrete-Stage Comparison · pp. 9–10
- 44.97% vs 35.73% mainlobe energy fraction and −3.60 vs −2.30 dB PSLL for Fluid+GS vs Fixed.Source: B. Discrete-Stage Comparison · p. 10
- At θ = 45°, Fluid+GS has the highest main peaks among the four discrete configurations; at φ = 30°, it has a more concentrated main beam and lower nearby sidelobe shoulders.Source: B. Discrete-Stage Comparison · p. 10, Figure 6 · p. 10
C. Continuous-Stage Comparison: Adam Refinement
Continuous Adam refinement improves the two-beam design, while the conclusion reports stronger mainlobe concentration and lower peak sidelobes than a fixed array with the same aperture and active-port count. The section also compares continuous-stage behavior across beam cross sections and tests the benefit of OMP initialization.
C. Continuous-Stage Comparison: Adam Refinement
- Fluid+GS+Adam starts from OMP-selected positions and least-squares weights, then jointly refines continuous port coordinates and complex weights with spacing repair after each position update.The refinement uses the concentration-and-balance loss.Source: C. Continuous-Stage Comparison: Adam… · p. 10
- 48.92% mainlobe energy fraction versus 44.97% for Fluid+GS, with PSLL reduced from −3.60 dB to −3.83 dB after Adam refinement.The mainlobe-energy increase is 3.95 percentage points over Fluid+GS.Source: C. Continuous-Stage Comparison: Adam… · p. 10
- At iteration 100, Fluid+GS initialization yields a 2.22-percentage-point higher mainlobe energy fraction than uniform-array initialization; at iteration 300, the advantage is 1.49 points.Fluid+GS initialization is higher at every listed checkpoint under the shared learning-rate schedule.Source: C. Continuous-Stage Comparison: Adam… · p. 10, Table III · p. 10
- At θ = 45◦, Adam produces the highest main peaks while maintaining two-beam balance; at φ = 30◦, it narrows nearby shoulders and reduces response outside the mainlobe relative to Fluid+GS.Source: C. Continuous-Stage Comparison: Adam… · p. 10
V. CONCLUSION
- The conclusion’s pipeline uses FFT-based Gerchberg–Saxton phase retrieval to construct an aperture-compatible complex target from an amplitude-only desired pattern.OMP selection and least squares provide the discrete synthesis and initial weights for continuous refinement.Source: §V · p. 11
- 35.73% to 48.92% mainlobe energy fraction and −2.30 dB to −3.83 dB PSLL versus the fixed array with the same aperture and active-port count.These are the complete method’s results in the evaluated two-beam setup.Source: §V · p. 11