OFDM Enabled Over-the-Air Computation Systems with Two-Dimensional Fluid Antennas

Fluid antenna system (FAS) is able to exploit spatial degrees of freedom (DoFs) in wireless channels. In this letter, to exploit spatial DoFs in frequency-selective environments, we investigate an orthogonal frequency division multiplexing enabled over-the-air computation system, where the access point is equipped with a two-dimensional FAS to enhance performance. We solve the computation mean square error (MSE) minimization problem by transforming the original problem into transmit precoders optimization problem and antenna positions optimization along with receive combiners optimization problem. The latter is solved via a majorization-minimization approach combined with sequential optimization. Numerical results confirm that the proposed scheme achieves MSE reduction over the scheme with fixed position antennas.

💡 Research Summary

This paper investigates the integration of a two‑dimensional fluid‑antenna system (FAS) with orthogonal frequency‑division multiplexing (OFDM) enabled over‑the‑air computation (AirComp). In a multi‑user wireless network, K single‑antenna devices transmit data simultaneously to an access point (AP) equipped with M fluid antennas that can move within a bounded planar region ℛ while maintaining a minimum inter‑antenna distance δ. The wideband channel is modeled as L multipath components per user, each characterized by a complex gain, delay, and elevation/azimuth angles. The spatial phase shift induced by antenna position is captured by a simple linear function of the coordinates, enabling a compact representation of the channel matrix as a product of a field‑response matrix F and a diagonal gain matrix G.

Each user applies a complex precoder b_{k,n} on subcarrier n, and the AP employs a linear combiner w_n to estimate the desired function Σ_k c_{k,n} (the sum of the transmitted symbols). The mean‑square error (MSE) of the AirComp estimate is derived in closed form (Eq. 14) and averaged over all N subcarriers (Eq. 15). The design goal is to minimize this overall MSE subject to per‑subcarrier power constraints |b_{k,n}|² ≤ P, antenna‑position constraints r_m ∈ ℛ, and the minimum distance constraint ‖r_m – r_{m’}‖ ≥ δ.

The resulting optimization problem is non‑convex because the precoders, combiners, and antenna positions are coupled in a highly nonlinear way. The authors decompose the problem into three sub‑problems using an alternating optimization (AO) framework:

1. Precoding sub‑problem – With w_n and r fixed, each b_{k,n} can be optimized independently. The solution is a simple water‑filling‑type expression that scales the matched‑filter term w_n^H h_{k,n}(r) to satisfy the power budget (Eq. 18).

2. Combiner sub‑problem – With b_{k,n} and r fixed, the optimal w_n is obtained by solving a regularized least‑squares problem, yielding a closed‑form expression (Eq. 20). Substituting this back produces an objective that depends only on the antenna positions.

3. Antenna‑position sub‑problem – This is the most challenging part because the channel vectors h_{k,n}(r) and the matrix V_n = H_n B_n H_n^H + σ²I both depend on r. To handle the non‑convexity, the authors adopt a majorization‑minimization (MM) approach. At iteration τ they fix H_n^{(τ)} and V_n^{(τ)} and construct a surrogate lower bound using the first‑order Taylor expansion of the trace terms (Eq. 35). Lemma 1 provides a matrix inequality that guarantees the surrogate is a global under‑estimator of the original objective.

The surrogate separates across antennas, leading to a per‑antenna maximization problem of the form
max_{r_m} –2 Re{ η(r_m)^H ϕ_m }
subject to r_m ∈ ℛ and the distance constraints. Here η(r_m) aggregates the channel responses of all users at antenna m, while ϕ_m aggregates the contributions from all subcarriers. Because the only coupling among antennas is the minimum‑distance constraint, the authors solve the positions sequentially: first optimize r_1 by exhaustive 2‑D grid search, then restrict the feasible region for r_2 by removing a disk of radius δ around r_1, and so on. This sequential search yields a tractable algorithm with complexity O(M·|grid|).

The complete AO‑MM algorithm proceeds as follows: initialize antenna positions and combiners, compute the optimal precoders, update the MM surrogate, perform the sequential 2‑D search for each antenna, recompute the combiners, and iterate until the MSE change falls below a threshold. In simulations the algorithm converges within 10–15 iterations.

Numerical Results – The authors evaluate a scenario with K = 5 users, N = 64 subcarriers, M = 4 fluid antennas, L = 4 paths, carrier frequency 2.4 GHz (λ = 0.125 m), and δ = λ/2. The feasible region spans ±1.5λ in both dimensions. Figure 1 shows MSE versus transmit‑to‑noise ratio (P/σ²) for four schemes: the proposed AO‑MM method, a fixed‑position antenna (FPA) baseline, an exhaustive‑search (EAS) benchmark that selects the best positions from a discrete set, and a successive convex approximation (SCA) method from prior work. The proposed scheme consistently outperforms the baselines, achieving up to 5 dB MSE reduction at high SNR. Figure 2 plots MSE versus the number of users K (P/σ² = 10 dB). All schemes degrade as K grows, but the proposed method maintains the lowest MSE across the entire range, confirming that the additional spatial degrees of freedom provided by the fluid antennas effectively mitigate multi‑user interference in the AirComp aggregation.

Conclusions – By jointly optimizing transmit precoders, receive combiners, and fluid‑antenna positions, the paper demonstrates substantial MSE improvements for OFDM‑based AirComp in frequency‑selective channels. The AO‑MM framework offers a practical balance between computational complexity and performance, while the sequential 2‑D search respects realistic hardware constraints such as minimum antenna spacing. Limitations include reliance on a grid‑based position search (which may become costly for finer resolutions or larger M) and the assumption of perfect channel state information. Future work could explore gradient‑based continuous position updates, robustness to CSI errors, and extensions to multi‑cell or massive‑MIMO settings.