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

TL;DR

The paper addresses the problem of designing low-complexity beam coverage for MIMO systems in both far-field and near-field regimes, aiming to maximize the worst-case beamforming gain over a prescribed target region under a transmit-power constraint. It introduces a Fourier-transform–based approach: in the far field, an inverse FT maps the angular coverage to an antenna-domain weight shaping with a real-valued, rectangular gain target; in the near field, a first-order Taylor expansion of the CSV enables a 2D inverse FT over angle and inverse range, yielding a separable 2D sinc-weight design. A roll-off–aware mechanism with a protective zoom is developed for finite antenna arrays, and a near-field range defocusing effect is revealed when angular coverage is wide. Numerical results show the FT-based designs achieve performance comparable to dense sampling methods but with orders-of-magnitude reduced computational complexity, enabling real-time beam training and tracking, and the framework is extensible to multi-region coverage and analog beamforming.

Abstract

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.

Figures (7)

• Figure 1: Beamforming gain and magnitude of $[\mathbf{w}_{\rm FF}]_{n}$.
• Figure 2: Unconstrained beamforming gain in the angle domain given $\theta_{\min} =-0.2$, $\theta_{\max} =0.2$, $f = 30$ GHz, and $N=64$.
• Figure 3: $\mathcal{L}(\Delta\theta, \Delta\xi)$ versus angle deviation and inverse range deviation when $N=256$, $f=30$ GHz, $\theta_0 = 0$, and $\xi_0 = 1/15$ with $[\theta_{\min}, \theta_{\max}]=[-\frac{\sqrt{3}}{2},\frac{\sqrt{3}}{2}]$ and $[\xi_{\min}, \xi_{\max}] = [ \frac{1}{Z_{\rm Rayl}}, \frac{1}{Z_{\rm Fres}}]$.
• Figure 4: Unconstrained beamforming gain in the range domain for the near-field case given inverse range deviation $\Delta \xi = 0$, reference point $(0,1/15)$, $N=256$, and $f=30$ GHz under different angle deviations $\mu$.
• Figure 5: Beam coverage design for the far-field case.

Theorems & Definitions (14)

• Lemma 1: Far-field CSV
• Lemma 2: FT between antenna domain and spatial frequency domain
• Example 1
• Lemma 3: Unconstrained beamforming gain
• proof
• Proposition 1: Roll-off property
• proof
• Example 2
• Lemma 4: Approximated near-field CSV
• proof