Low-complexity Design for Beam Coverage in Near-field and Far-field: A Fourier Transform Approach

In this paper, we study efficient beam coverage design for multi-antenna systems in both far-field and near-field cases. To reduce the computational complexity of existing sampling-based optimization methods, we propose a new low-complexity yet efficient beam coverage design. To this end, we first formulate a general beam coverage optimization problem to maximize the worst-case beamforming gain over a target region. For the far-field case, we show that the beam coverage design can be viewed as a spatial-frequency filtering problem, where angular coverage can be achieved by weight-shaping in the antenna domain via an inverse FT, yielding an infinite-length weighting sequence. Under the constraint of a finite number of antennas, a surrogate scheme is proposed by directly truncating this sequence, which inevitably introduces a roll-off effect at the angular boundaries, yielding degraded worst-case beamforming gain. To address this issue, we characterize the finite-antenna-induced roll-off effect, based on which a roll-off-aware design with a protective zoom is developed to ensure a flat beamforming-gain profile within the target angular region. Next, we extend the proposed method to the near-field case. Specifically, by applying a first-order Taylor approximation to the near-field channel steering vector (CSV), the two-dimensional (2D) beam coverage design (in both angle and inverse-range) can be transformed into a 2D inverse FT, leading to a low-complexity beamforming design. Furthermore, an inherent near-field range defocusing effect is observed, indicating that sufficiently wide angular coverage results in range-insensitive beam steering. Finally, numerical results demonstrate that the proposed FT-based approach achieves a comparable worst-case beamforming performance with that of conventional sampling-based optimization methods while significantly reducing the computational complexity.

💡 Research Summary

This paper addresses the challenging problem of designing beam coverage for large‑scale antenna arrays in both far‑field (planar wavefront) and near‑field (spherical wavefront) regimes. The authors formulate a worst‑case beam‑forming gain maximization problem over a prescribed spatial region—an angular interval for far‑field and a two‑dimensional (angle, inverse‑range) region for near‑field. Traditional approaches rely on dense angular sampling and iterative convex approximations, leading to computational complexities that scale linearly with both the number of antennas and the number of sampled points, which becomes prohibitive for XL‑MIMO systems.

The key insight of the work is to reinterpret the beam‑coverage design as a Fourier‑transform (FT) relationship between the antenna‑domain weighting vector and the spatial‑domain beam pattern. In the far‑field case, the ideal beam pattern over a target angular interval corresponds to a rectangular spectrum. Its inverse FT yields an infinite‑length weighting sequence (a sinc‑type function). Since a practical array has only N elements, the sequence must be truncated, which introduces a roll‑off (edge attenuation) at the angular boundaries and degrades the worst‑case gain. To mitigate this, the authors analytically characterize the roll‑off effect and introduce a “protective zoom” factor that slightly shrinks the target interval in proportion to 1/N before truncation. This pre‑emptive narrowing prevents the roll‑off from entering the region of interest, resulting in a flat gain profile across the desired angles while still using only N antennas.

For the near‑field case, the channel steering vector (CSV) contains a quadratic phase term due to spherical wavefronts. By applying a first‑order Taylor expansion to the phase with respect to the angle (assuming moderate angular deviations), the phase becomes approximately linear in both the antenna index and the inverse range ξ = 1/r. Consequently, the two‑dimensional beam‑coverage problem (over angle and ξ) can be expressed as a 2‑D inverse FT of a rectangular window in the (θ, ξ) domain. The resulting weighting matrix is a 2‑D sinc function, again truncated to the available N × N elements. An important observation is the “range defocusing” effect: when the angular coverage is sufficiently wide, the ξ‑dependent modulation averages out, and the beam becomes insensitive to range, effectively providing uniform coverage over a broad distance interval without additional range‑specific weighting.

Complexity analysis shows that the proposed FT‑based designs require only FFT/IFFT operations, yielding O(N log N) computational cost, compared with O(N·|Θ|·|Ξ|) for conventional sampling‑based methods. Numerical simulations confirm that the worst‑case gain of the FT approach matches that of high‑complexity benchmarks (within 0.5 dB) while reducing runtime by several orders of magnitude (e.g., from 885 ms to 0.0074 ms in far‑field and from 1275 ms to 0.0292 ms in near‑field).

The paper also discusses extensions to uniform planar arrays, multi‑region coverage, large angular deviations (requiring higher‑order Taylor terms), and analog phase‑only beamforming architectures. Overall, the work introduces a principled, low‑complexity framework that transforms beam‑coverage design into a spectral shaping problem, offering near‑optimal performance with dramatically reduced computational burden—making it highly suitable for real‑time beam training, tracking, and other latency‑critical operations in future 6G and XL‑MIMO systems.