Bessel Beam Optimization for Near-Field THz Communications under UE Location Uncertainty

Technical Summary: Bessel Beam Optimization for Near-Field THz Communications under UE Location Uncertainty

Problem Statement
Future Terahertz (THz) wireless systems rely on extremely large antenna arrays to overcome severe propagation losses and achieve terabit-per-second data rates. However, the combination of short wavelengths and large array apertures extends the radiative near-field zone to hundreds of meters, rendering conventional far-field plane-wave assumptions invalid. In this near-field regime, User Equipment (UE) often operates where beam focusing is required. While Bessel beams offer distinct advantages for mobile THz links—specifically their "non-diffracting" behavior (sustaining axial power over a range) and "self-healing" capability (resilience to blockage)—optimizing their configuration remains challenging.

Unlike far-field beamforming (determined by angle) or standard near-field focusing (determined by angle and distance), Bessel beams introduce an additional design parameter: the "Bessel cone angle" ($\theta$). Existing approaches either rely on heuristic parameter choices, which may be suboptimal, or perform exhaustive searches over configurations, which are computationally expensive and unsuitable for real-time adaptation. Furthermore, practical UE location estimates are imperfect. The optimal Bessel beam configuration depends jointly on the estimated location, the uncertainty distribution, and the performance metric, making real-time optimization difficult with existing methods.

Methodology
The authors formulate the problem as maximizing the expected Spectral Efficiency (SE) of a single-user downlink THz Multiple-Input Multiple-Output (MIMO) link under imperfect UE location knowledge. The system model considers a Uniform Planar Array (UPA) at the Access Point (AP) and a UPA at the UE, operating in the radiative near-field region with a line-of-sight (LoS) non-uniform spherical-wave channel.

The study addresses three specific UE location scenarios:

1. Deterministic: Perfect location knowledge ($\epsilon_z = 0$).
2. Gaussian Uncertainty: Location errors following a normal distribution ($\epsilon_z \sim \mathcal{N}(0, \sigma_G^2)$).
3. Uniform Uncertainty: Location errors following a uniform distribution ($\epsilon_z \sim \mathcal{U}(-\delta_U, \delta_U)$).

To solve the optimization problem without exhaustive search, the authors derive low-complexity closed-form approximations:

• Deterministic Case: Using a stationary-phase approximation and geometric scaling arguments, the authors model the AP elements as concentric rings. They derive that the dominant axial power peak occurs at $z_{peak} = Z_{max}/2$, where $Z_{max}$ is the maximum non-diffracting range. The optimal cone angle is determined by aligning this peak with the estimated UE location ($\mu$).
• Uncertain Cases: For Gaussian and uniform uncertainties, the authors extend the peak-alignment principle. They introduce first-order correction factors ($\alpha$ and $\beta$) to shift the target alignment point inward from the mean location $\mu$ to account for the uncertainty distribution. The optimal angle is found by aligning the beam's dominant peak with $\mu - \alpha\sigma_G$ (Gaussian) or $\mu - \beta\sigma_U$ (Uniform).

The correction coefficients $\alpha$ and $\beta$ are calibrated against an exhaustive search benchmark using numerical simulations across various array sizes and frequencies.

Key Contributions and Results

1. Closed-Form Solutions: The paper provides explicit, $O(1)$ complexity formulas for the optimal Bessel cone angle under deterministic, Gaussian, and uniform location uncertainty.

   • Deterministic: $\theta^* = \tan^{-1}\left(\frac{D_{AP}}{2\sqrt{2}\mu}\right)$
   • Gaussian: $\theta^* = \tan^{-1}\left(\frac{D_{AP}}{2\sqrt{2}(\mu - 0.22\sigma_G)}\right)$
   • Uniform: $\theta^* = \tan^{-1}\left(\frac{D_{AP}}{2\sqrt{2}(\mu - 0.40\sigma_U)}\right)$
     (Where $D_{AP}$ is the side length of the AP array).

2. Validation: Extensive simulations across frequencies (100–1000 GHz), array sizes (300–800 elements), and uncertainty levels demonstrate that these closed-form approximations closely match the performance of an exhaustive search.

   • The difference in Spectral Efficiency between the proposed method and the exhaustive search benchmark is negligible, with maximum 95th-percentile errors of 0.057% (deterministic), 0.00722% (Gaussian), and 0.00471% (uniform).
   • The proposed configurations track the optimal cone angle $\theta^*$ with high fidelity.

3. Performance Insights:

   • Uncertainty Impact: Under location uncertainty, the optimal strategy involves an "inward correction," shifting the beam's peak closer to the transmitter to maximize expected SE.
   • Robustness: The performance gap between deterministic and uncertain cases shrinks as the UE distance increases, but widens as the magnitude of location uncertainty grows.
   • Array Gain: Increasing the number of antenna elements ($N_{AP}$ and $N_{UE}$) consistently improves SE, while increasing frequency (with fixed physical aperture) slightly reduces SE due to reduced array gain and tighter axial regions.

Significance and Claims
The authors claim that their work addresses a critical gap in the design of practical near-field THz systems by providing an efficient, analytically tractable method for Bessel beam configuration. The primary significance lies in reducing the configuration complexity from a computationally prohibitive exhaustive search to a simple closed-form expression ($O(1)$) without sacrificing performance.

The paper asserts that these approximations are robust across a wide range of system parameters and uncertainty regimes. Consequently, the derived expressions facilitate the design of future near-field THz systems, including codebook design, and enable accurate, analytically tractable system-level studies regarding capacity, coverage, and interference modeling. The authors maintain a modest stance, noting that while the approximations are highly accurate, they rely on first-order geometric arguments and empirical calibration, leaving higher-order residual effects as minor deviations.